  A slope $\frac pq$ is called a characterizing slope for a given knot
$K_0\subset S^3$ if whenever the $\frac pq$--surgery on a knot $K\subset S^3$
is homeomorphic to the $\frac pq$--surgery on $K_0$ via an orientation
preserving homeomorphism, then $K=K_0$. In a previous paper, we showed that,
outside a certain finite set of slopes, only the negative integers could
possibly be non-characterizing slopes for the torus knot $T_{5,2}$. Applying
recent work of Baldwin--Hu--Sivek, we improve our result by showing that a
nontrivial slope $\frac pq$ is a characterizing slope for $T_{5,2}$ if $\frac
pq>-1$ and $\frac pq\notin \{0,1, \pm\frac12,\pm\frac13\}$. In particular,
every nontrivial L-space slope of $T_{5,2}$ is characterizing for $T_{5,2}$. As
a consequence, if a nontrivial $\frac pq$-surgery on a non-torus knot in $S^3$
yields a manifold of finite fundamental group, then $|p|>9$.
