Given an undirected graph $G=(V,E)$ and an integer $\ell$, the Eccentricity Shortest Path (ESP) asks to find a shortest path $P$ such that for every vertex $v\in V(G)$, there is a vertex $w\in P$ such that $d_G(v,w)\leq \ell$, where $d_G(v,w)$ represents the distance between $v$ and $w$ in $G$. Dragan and Leitert [Theor. Comput. Sci. 2017] showed that the optimization version of this problem, which asks to find the minimum $\ell$ for the ESP problem, is NP-hard even on planar bipartite graphs with maximum degree 3. They also showed that ESP is W[2]-hard when parameterized by $\ell$. On the positive side, Ku\v cera and Suchý [IWOCA 2021] showed that the problem exhibits fixed parameter tractable (FPT) behavior when parameterized by modular width, cluster vertex deletion set, maximum leaf number, or the combined parameters disjoint paths deletion set and $\ell$. It was asked as an open question in the above paper, if ESP is FPT parameterized by disjoint paths deletion set or feedback vertex set. We answer these questions partially and obtain the following results: - ESP is FPT when parameterized by disjoint paths deletion set, split vertex deletion set or the combined parameters feedback vertex set and eccentricity of the graph. - We design a $(1+ε)$-factor FPT approximation algorithm when parameterized by the feedback vertex set number. - ESP is W[2]-hard when parameterized by the chordal vertex deletion set.