The answer is yes: $\mathbf{Pic}_{X/S}$ is representable by a scheme. I will argue that this follows from the SGA 3 result mentioned by user27920 and from Theorem 2 (c) in section 6.6 of *Neron models* (which itself is based on a nonflat descent result due to Raynaud).

**Preliminary reductions:** Since the $g \neq 1$ case is clear, let us assume that $g = 1$. It suffices to work locally on $S$, so standard limit arguments reduce to the case when $S$ is Noetherian and affine. Since $\mathbf{Pic}_{X/S} = \bigsqcup_n \mathbf{Pic}^n_{X/S}$, it suffices to show each $\mathbf{Pic}^n_{X/S}$ is a scheme. The latter is a torsor under the elliptic curve $E:=\mathbf{Pic}^0_{X/S}$, so is an $S$-algebraic space of finite presentation. Further limit arguments therefore reduce to the case when $S$ is in addition local.

**The main argument:** We will argue that every finite set of points of $X$ is contained in an open affine. As explained by user27920, it will then follow from SGA 3, V.4.1 that each $\mathbf{Pic}^n_{X/S}$ is a scheme.

Choose an affine open $Y \subset X$ that meets the closed fiber of $X \rightarrow S$. Since $X \rightarrow S$ is open and $S$ is local, $Y$ automatically meets every fiber. The action map $E \times_S X \rightarrow X$ (gotten from the identification $X = \mathbf{Pic}^1_{X/S}$) is open because it is a base change of $E \rightarrow S$, so the image $EY$ of $E \times_S Y$ is open. Then checking fiberwise, we conclude that $EY = X$. Therefore, the claim about points being in an open affine follows from the result from *Neron models* mentioned earlier, a simplified version of which says:

**Theorem.** Let $S$ be an affine Noetherian scheme, let $E$ be a smooth $S$-group scheme with connected fibers, and let $E \times_S X \rightarrow X$ be a group action of $E$ on a smooth $S$-scheme $X$. If there is an affine open subscheme $Y \subset X$ such that $EY = X$, then any finite set of points of $X$ is contained in an open affine.

canonicallyprojective, so by working etale-locally on the base to attain projectivity we see in general that Pic$_{X/S}$ is an fppf sheaf extension $P$ of $\mathbf{Z}$ by an elliptic curve $E$. Let $P_n$ be the fiber over $n$, so $P_1$ is the curve $X$ that is an $E$-torsor. For $n\ne 0$, $[n]:P \rightarrow P$ expresses $P_n$ as the quotient of $P_1=X$ by the free action of the finite loc. free $E[n]$. So equivalently: is the algebraic space $X/E[n]$ a scheme for $n\ne 0$? By SGA3, Exp. V, Thm. 4.1, same that each fibral $E[n]$-orbit is in an open affine. $\endgroup$Zariski-locallyprojective, which suffices to show Pic is a scheme under those assumptions. Indeed, let $s\in S$ be a point and $U\subset X$ an affine in $X$ such that $f^{-1}(S)\cap U$ is non-empty. Then by normality the complement $D$ of $U$ has pure codimension $1$; by local factoriality it is a Cartier divisor on $X$, and by non-emptyness of $f^{-1}(S)\cap U$ it is relatively ample over a neighborhood of $s$. (Note that $D\cap f^{-1}(s)$ is non-empty as $U$ is affine.) $\endgroup$normal, then $X=\underline{\mathrm{Pic}}^1_{X/S}$ is locally projective over $S$ by the main theorem in the introduction to Raynaud's thesis (LNM 119, also Corollaire VI.2.5). By the first comment of user52824, it follows that $\underline{\mathrm{Pic}}_{X/S}$ is a scheme (in fact each $\underline{\mathrm{Pic}}^n_{X/S}$ is locally projective over $S$). $\endgroup$2more comments