  We study pseudo-geometric strongly regular graphs whose second subconstituent
with respect to a vertex is a cover of a strongly regular graph or a complete
graph. By studying the structure of such graphs, we characterize all graphs
containing such a vertex, and use our characterization to find many new
strongly regular graphs. Thereby, we answer a question posed by Gardiner,
Godsil, Hensel, and Royle. We give an explicit construction for q new, pairwise
non-isomorphic graphs with the same parameters as the collinearity graph of
generalized quadrangles of order $(q,q)$ and a new non-geometric graph with the
same parameters as the collinearity graph of the Hermitian generalized
quadrangle of order $(q^2, q)$, for prime powers $q$. Using our
characterization, we computed 135478 new strongly regular graphs with
parameters (85,20,3,5) and 27 039 strongly regular graphs with parameters (156,
30, 4, 6).
