  Let $F$ be a non-archimedean local field of characteristic different from $2$
and $G$ be either an odd special orthogonal group ${\rm SO}_{2r+1}(F)$ or a
symplectic group ${\rm Sp}_{2r}(F)$. In this paper, we establish the local
converse theorem for $G$. Namely, for given two irreducible admissible generic
representations of $G$ with the same central character, if they have the same
local gamma factors twisted by irreducible supercuspidal representations of
${\rm GL}_n(F)$ for all $1 \leq n \leq r$ with the same additive character,
these representations are isomorphic. Using the theory of Cogdell, Shahidi, and
Tsai on partial Bessel functions and the classification of irreducible generic
representations, we break the barrier on the rank of twists $1 \leq n \leq
2r-1$ in the work of Jiang and Soudry, and extend the result of Q. Zhang, which
was achieved for all supercuspidal representations in characteristic $0$.
