TLDR: I want to know if there's a simple way to fill in distances for all vertices reachable from negative weight cycles (not just ones on the cycle itself) once Bellman-Ford has found a negative-weight cycle.
The Bellman-Ford algorithm finds single-source shortest distances $d$ for a graph of $|V|$ vertices by making $|V|-1$ passes through all $|E|$ edges with weights $w$, and then checking each edge $u \rightarrow v$, with weight $w(u,v)$, for negative-weight cycles. If $v.d > u.d + w(u,v)$, then we know that there exists a negative weight cycle consisting of some path from $v$ to $u$, then the edge $(u,v)$. It's straightforward to find every vertex on that cycle and set all their $d$ to $-\infty$.
However, this also must mean that any vertices reachable from but not on this cycle must also have $d = -\infty$. One way to update all their distances is to collapse each cycle to a vertex and then perform DFS from each such "cycle vertex", setting $x.d = -\infty$ for every reachable vertex $x$.
I'm wondering if there's a faster way to update all the distances, involving perhaps a smaller modification to Bellman-Ford?