A partial answer for transpositions:
If $A$ contains a generating set given by transpositions, there is an $n-$cycle given by the product of $n-1$ generators. Indeed, consider the graph $G$ with vertices $1,\dots,n$ and edges defined by all transpositions in the generating set $A$. The subset of transpositions in $A$ generates $S_n$ if and only if $G$ is a connected graph. Up to throwing away a few generators, we can thus assume that $G$ is a tree. Drawing this tree in the plane and choosing an initial leaf of $G$, we consider the product of all transpositions corresponding to the order in which we encounter the edges of $G$ when walking around $G$ in counterclockwise order. This defines the $n-$cycle corresponding to the inverse of the cycle obtained by writing down all vertices of $G$ accordingly to their first last sighting during the walk.
