We make progress toward a characterization of the graphs $H$ such that every connected $H$-free graph has a longest path transversal of size $1$. In particular, we show that the graphs $H$ on at most $4$ vertices satisfying this property are exactly the linear forests. We also show that if the order of a connected graph $G$ is large relative to its connectivity $κ(G)$, and its independence number $α(G)$ satisfies $α(G) \le κ(G) + 2$, then each vertex of maximum degree forms a longest path transversal of size $1$.