  Let $F$ be a non-Archimedean local field with odd characteristic $p$. Let $N$
be a positive integer and $G=Sp_{2N}(F)$. By work of Lomel\'i on
$\gamma$-factors of pairs and converse theorems, a generic supercuspidal
representation $\pi$ of $G$ has a transfer to a smooth irreducible
representation $\Pi_\pi$ of $GL_{2N+1}(F)$. In turn the Weil-Deligne
representation $\Sigma_\pi$ associated to $\Pi_\pi$ by the Langlands
correspondence determines a Langlands parameter $\phi_\pi$ for $\pi$. That
process produces a Langlands correspondence for generic cuspidal
representations of $G$.
  In this paper we take $\pi$ to be simple in the sense of Gross and Reeder,
and from the explicit construction of $\pi$ we describe $\Pi_\pi$ explicitly.
The method we use is the same as in our previous paper arXiv:2310.20455, where
we treated the case where $F$ is a $p$-adic field, and $\pi$ a simple
supercuspidal representation of $G=Sp_{2N}(F)$. It relies on a criterion due to
Moeglin on the reducibility of representations parabolically induced from
$GL_M(F)\times G$ for varying positive integers $M$. We extend this criterion
to the case when $F$ has any positive characteristic. The main new feature
consists in relating reducibility to $\gamma$-factors for pairs.
