[Example 1 in A.Neumaier paper](https://doi.org/10.1016/S0304-0208(08)73275-4) says in partcular that the vertex degree in this case must be $k=s^2+1$, for $s$ not divisible by 4. As well, the number of vertices is $v=1+k+\binom{k}{2}$. The examples you list correspond to $s=2,3$.  The next possible parameter set corresponds to $s=5$, so you have $v=352$, $k=26$. [A.Brouwer's database](https://www.win.tue.nl/~aeb/graphs/srg/srgtab351-400.html) lists this tuple of parameters as feasible, but no examples known. Similarly for $s=6,7$ you have feasible sets of parameters $v=704,1276$, resp. $k=37,50$, but no examples known.