Clearly every generously transitive permutation group $G < {\rm S}_n$ acts primitively on $\{1,\dots,n\}$. Therefore we can find all generously transitive permutation groups of degree $n$ by searching through the primitive groups of degree $n$. The latter are available in the Primitive Permutation Groups Library of GAP, for all $n \leq 2500$. The following GAP function tests whether a given group is generously transitive:

IsGenerouslyTransitive := function ( G )

  local  n, i, j;

  n := LargestMovedPoint(G);
  if   not IsTransitive(G,[1..n]) or not IsPrimitive(G,[1..n])
  then return false; fi;
  if Transitivity(G,[1..n]) >= 2 then return true; fi;
  for i in [1..n] do
    for j in [i+1..n] do
      if   RepresentativeAction(G,[i,j],[j,i],OnTuples) = fail
      then return false; fi;
    od;
  od;
  return true;
end;

Now let's check how many of the primitive groups of degree $\leq 100$ are even generously transitive:

gap> List([1..100],NrPrimitiveGroups); 
[ 0, 1, 2, 2, 5, 4, 7, 7, 11, 9, 8, 6, 9, 4, 6, 22, 10, 4, 8, 4, 9, 4, 
   7, 5, 28, 7, 15, 14, 8, 4, 12, 7, 4, 2, 6, 22, 11, 4, 2, 8, 10, 4, 10, 
  4, 9, 2, 6, 4, 40, 9, 2, 3, 8, 4, 8, 9, 5, 2, 6, 9, 14, 4, 8, 74, 13, 
  7, 10, 7, 2, 2, 10, 4, 16, 4, 2, 2, 4, 6, 10, 4, 155, 10, 6, 6, 6, 2, 
  2, 2, 10, 4, 10, 2, 2, 2, 2, 2, 14, 4, 2, 38 ]
gap> Sum(last); # total number of primitive groups of degree <= 100
946
gap> List([1..100],n->Number(AllPrimitiveGroups(DegreeAction,n),
>                            IsGenerouslyTransitive));
[ 0, 1, 1, 2, 4, 4, 5, 7, 11, 9, 6, 6, 7, 4, 6, 22, 9, 4, 5, 4, 9, 4, 5, 
  5, 26, 7, 11, 14, 6, 4, 8, 7, 4, 2, 6, 21, 8, 4, 2, 8, 8, 4, 6, 4, 9, 
  2, 4, 4, 38, 9, 2, 3, 6, 4, 7, 9, 5, 2, 4, 9, 10, 4, 7, 74, 13, 7, 6, 
  7, 2, 2, 6, 4, 13, 4, 2, 2, 4, 5, 6, 4, 150, 10, 4, 6, 6, 2, 2, 2, 8, 
  4, 8, 2, 2, 2, 2, 2, 12, 4, 2, 38 ]
gap> Sum(last); # number of generously transitive groups among them
867

Hence over $90$ percent of the primitive groups of degree $\leq 100$ are even generously transitive. -- Therefore, perhaps rather than asking for a classification of generously transitive groups, one might ask for a classification of those primitive permutation groups which are not generously transitive.

Clearly every generously transitive permutation group $G < {\rm S}_n$ acts transitively on $\{1,\dots,n\}$. Therefore we can find all generously transitive permutation groups of degree $n$ by searching through the transitive groups of degree $n$. The latter are available in the Transitive Permutation Groups Library of GAP, for all $n \leq 30$. The following GAP function tests whether a given group is generously transitive:

IsGenerouslyTransitive := function ( G )

  local  n, i, j;

  n := LargestMovedPoint(G);
  if not IsTransitive(G,[1..n]) then return false; fi;
  if Transitivity(G,[1..n]) >= 2 then return true; fi;
  for i in [1..n] do
    for j in [i+1..n] do
      if   RepresentativeAction(G,[i,j],[j,i],OnTuples) = fail
      then return false; fi;
    od;
  od;
  return true;
end;

Now let's check how many of the transitive groups of degree $\leq 20$ are even generously transitive:

gap> List([2..20],NrTransitiveGroups);                                         
[ 1, 2, 5, 5, 16, 7, 50, 34, 45, 8, 301, 9, 63, 104, 1954, 10, 983, 8, 1117 ]
gap> Sum(last);
4722
gap> List([2..20],n->Number(AllTransitiveGroups(DegreeAction,n),
>                           IsGenerouslyTransitive));
[ 1, 1, 4, 4, 11, 5, 37, 22, 39, 6, 191, 7, 47, 63, 1239, 9, 605, 5, 870 ]
gap> Sum(last);
3166

Hence about two thirds of the transitive groups of degree $\leq 20$ are even generously transitive. -- Therefore, perhaps rather than asking for a classification of generously transitive groups, one might ask for a classification of those transitive permutation groups which are not generously transitive.

Clearly every generously transitive permutation group $G < {\rm S}_n$ acts primitively on the set $\{1,\dots,n\}$. Therefore we can find all generously transitive permutation groups of degree $n$ by searching through the primitive groups of degree $n$. The latter are available in the Primitive Groups Library of the computer algebra system GAP (cf. http://www.gap-system.org/Datalib/prim.html), for all  $n \leq 2500$. First we write a GAP function which tests whether a given group is generously transitive:

Then we apply this function to the groups in the Primitive Groups Library. Since it turns out that most primitive groups of small degree are even generously transitive, in order to avoid the output getting too long, we list those which are not generously transitive:

gap> NotGenerouslyTransitivePrimitiveGroupsOfDegree :=
>      n -> Filtered( AllPrimitiveGroups( DegreeAction, n ),
>                     G -> not IsGenerouslyTransitive( G ) );;
gap> for n in [1..150] do
>      grps := NotGenerouslyTransitivePrimitiveGroupsOfDegree(n);
>      if grps <> [] then
>        Print("    The following primitive groups of degree ",n,
>              " are not generously transitive:\n",grps,"\n\n");
>      fi;
>    od;
The following primitive groups of degree 3 are not generously transitive:
[ A(3) ]

The following primitive groups of degree 5 are not generously transitive:
[ C(5) ]

The following primitive groups of degree 7 are not generously transitive:
[ C(7), 7:3 ]

The following primitive groups of degree 11 are not generously transitive:
[ C(11), 11:5 ]

The following primitive groups of degree 13 are not generously transitive:
[ C(13), 13:3 ]

The following primitive groups of degree 17 are not generously transitive:
[ C(17) ]

The following primitive groups of degree 19 are not generously transitive:
[ C(19), 19:3, 19:9 ]

The following primitive groups of degree 23 are not generously transitive:
[ C(23), 23:11 ]

The following primitive groups of degree 25 are not generously transitive:
[ 5^2:3, 5^2:S(3) ]

The following primitive groups of degree 27 are not generously transitive:
[ 3^3.A(4), 3^3:13, 3^3.S(4), 3^3.13.3 ]

The following primitive groups of degree 29 are not generously transitive:
[ C(29), 29:7 ]

The following primitive groups of degree 31 are not generously transitive:
[ C(31), 31:3, 31:5, 31:15 ]

The following primitive groups of degree 36 are not generously transitive:
[ PSU(3, 3) ]

The following primitive groups of degree 37 are not generously transitive:
[ C(37), 37:3, 37:9 ]

The following primitive groups of degree 41 are not generously transitive:
[ C(41), 41:5 ]

The following primitive groups of degree 43 are not generously transitive:
[ C(43), 43:3, 43:7, 43:21 ]

The following primitive groups of degree 47 are not generously transitive:
[ C(47), 47:23 ]

The following primitive groups of degree 49 are not generously transitive:
[ 7^2:S(3), 7^2:3 x D(2*3) ]

The following primitive groups of degree 53 are not generously transitive:
[ C(53), 53:13 ]

The following primitive groups of degree 55 are not generously transitive:
[ PSL(2, 11) ]

The following primitive groups of degree 59 are not generously transitive:
[ C(59), 59:29 ]

The following primitive groups of degree 61 are not generously transitive:
[ C(61), 61:3, 61:5, 61:15 ]

The following primitive groups of degree 63 are not generously transitive:
[ PSU(3, 3) ]

The following primitive groups of degree 67 are not generously transitive:
[ C(67), 67:3, 67:11, 67:33 ]

The following primitive groups of degree 71 are not generously transitive:
[ C(71), 71:5, 71:7, 71:35 ]

The following primitive groups of degree 73 are not generously transitive:
[ C(73), 73:3, 73:9 ]

The following primitive groups of degree 78 are not generously transitive:
[ PSL(2, 13) ]

The following primitive groups of degree 79 are not generously transitive:
[ C(79), 79:3, 79:13, 79:39 ]

The following primitive groups of degree 81 are not generously transitive:
[ 3^4:5, 3^4:D_10, 3^4:5:4, 3^4:Alt(5), 3^4:Sym(5) ]

The following primitive groups of degree 83 are not generously transitive:
[ C(83), 83:41 ]

The following primitive groups of degree 89 are not generously transitive:
[ C(89), 89:11 ]

The following primitive groups of degree 91 are not generously transitive:
[ PSL(2, 13), PSL(2, 13) ]

The following primitive groups of degree 97 are not generously transitive:
[ C(97), 97:3 ]

The following primitive groups of degree 101 are not generously transitive:
[ C(101), 101:5, 101:25 ]

The following primitive groups of degree 103 are not generously transitive:
[ C(103), 103:3, 103:17, 103:51 ]

The following primitive groups of degree 107 are not generously transitive:
[ C(107), 107:53 ]

The following primitive groups of degree 109 are not generously transitive:
[ C(109), 109:3, 109:9, 109:27 ]

The following primitive groups of degree 113 are not generously transitive:
[ C(113), 113:7 ]

The following primitive groups of degree 121 are not generously transitive:
[ 11^2:3, 11^2:D_6, 11^2:D_10, 11^2:15, 11^2:(5 x D_6), 11^2:(5 x D_10) ]

The following primitive groups of degree 125 are not generously transitive:
[ 5^3:Alt(4), 5^3:Sym(4), 5^3:31, 5^3:4^2:3, 5^3:31:3, 5^3:4^2:Sym(3) ]

The following primitive groups of degree 127 are not generously transitive:
[ C(127), 127:3, 127:7, 127:9, 127:21, 127:63 ]

The following primitive groups of degree 131 are not generously transitive:
[ C(131), 131:5, 131:13, 131:65 ]

The following primitive groups of degree 136 are not generously transitive:
[ PSL(2, 17) ]

The following primitive groups of degree 137 are not generously transitive:
[ C(137), 137:17 ]

The following primitive groups of degree 139 are not generously transitive:
[ C(139), 139:3, 139:23, 139:69 ]

The following primitive groups of degree 144 are not generously transitive:
[ PSL(3, 3), M_12 ]
 
The following primitive groups of degree 149 are not generously transitive:
[ C(149), 149:37 ]

Hence 1230 of the 1357 primitive groups of degree $\leq 150$ are even generously transitive, and only 127 of them are not.

Just as it is probably not reasonable to ask for a classification of all finite primitive groups up to conjugacy, this coincidence suggests that the same is the case for the question for a classification of all finite generously transitive groups. If one is lucky, maybe one can classify primitive groups which are  not generously transitive. -- This would give a classification of generously transitive groups in the sense that one accepts the primitive groups as known, and classifies the generously transitive groups as a subclass of them.

Clearly every generously transitive permutation group $G < {\rm S}_n$ acts primitively on $\{1,\dots,n\}$. Therefore we can find all generously transitive permutation groups of degree $n$ by searching through the primitive groups of degree $n$. The latter are available in the Primitive Permutation Groups Library of GAP, for all  $n \leq 2500$. The following GAP function tests whether a given group is generously transitive:

Now let's check how many of the primitive groups of degree $\leq 100$ are even generously transitive:

gap> List([1..100],NrPrimitiveGroups); 
[ 0, 1, 2, 2, 5, 4, 7, 7, 11, 9, 8, 6, 9, 4, 6, 22, 10, 4, 8, 4, 9, 4, 
  7, 5, 28, 7, 15, 14, 8, 4, 12, 7, 4, 2, 6, 22, 11, 4, 2, 8, 10, 4, 10, 
  4, 9, 2, 6, 4, 40, 9, 2, 3, 8, 4, 8, 9, 5, 2, 6, 9, 14, 4, 8, 74, 13, 
  7, 10, 7, 2, 2, 10, 4, 16, 4, 2, 2, 4, 6, 10, 4, 155, 10, 6, 6, 6, 2, 
  2, 2, 10, 4, 10, 2, 2, 2, 2, 2, 14, 4, 2, 38 ]
gap> Sum(last); # total number of primitive groups of degree <= 100
946
gap> List([1..100],n->Number(AllPrimitiveGroups(DegreeAction,n),
>                            IsGenerouslyTransitive));
[ 0, 1, 1, 2, 4, 4, 5, 7, 11, 9, 6, 6, 7, 4, 6, 22, 9, 4, 5, 4, 9, 4, 5, 
  5, 26, 7, 11, 14, 6, 4, 8, 7, 4, 2, 6, 21, 8, 4, 2, 8, 8, 4, 6, 4, 9, 
  2, 4, 4, 38, 9, 2, 3, 6, 4, 7, 9, 5, 2, 4, 9, 10, 4, 7, 74, 13, 7, 6, 
  7, 2, 2, 6, 4, 13, 4, 2, 2, 4, 5, 6, 4, 150, 10, 4, 6, 6, 2, 2, 2, 8, 
  4, 8, 2, 2, 2, 2, 2, 12, 4, 2, 38 ]
gap> Sum(last); # number of generously transitive groups among them
867

Hence over $90$ percent of the primitive groups of degree $\leq 100$ are even generously transitive. -- Therefore, perhaps rather than asking for a classification of generously transitive groups, one might ask for a classification of those primitive permutation groups which are  not generously transitive.