The path eccentricity of a connected graph $G$ is the minimum integer $k$ such that $G$ has a path such that every vertex is at distance at most $k$ from the path. A result of Duffus, Jacobson, and Gould from 1981 states that every connected $\{\text{claw}, \text{net}\}$-free graph $G$ has a Hamiltonian path, that is, $G$ has path eccentricity $0$. Several more recent works identified various classes of connected graphs with path eccentricity at most $1$, or, equivalently, graphs having a spanning caterpillar, including connected $P_5$-free graphs, AT-free graphs, and biconvex graphs. Generalizing all these results, we apply the work on structural distance domination of Bacsó and Tuza [Discrete Math., 2012] and characterize, for every positive integer $k$, graphs such that every connected induced subgraph has path eccentricity less than $k$. More specifically, we show that every connected $\{S_{k}, T_{k}\}$-free graph has a path eccentricity less than $k$, where $S_k$ and $T_k$ are two specific graphs of path eccentricity $k$ (a subdivided claw and the line graph of such a graph). As a consequence, every connected $H$-free graph has path eccentricity less than $k$ if and only if $H$ is an induced subgraph of $3P_{k}$ or $P_{2k+1} + P_{k-1}$. For such cases, we also provide a robust polynomial-time algorithm that finds a path witnessing the upper bound on the path eccentricity. Our main result also answers an open question of Bastide, Hilaire, and Robinson [Discrete Math., 2025].