If I understand your question correctly, this is true. Let me be a little careful about terminology, because there are multiple notions of lift, but luckily in this case they all agree.
When talking about lifts, one of the most general things you can ask is a flat proper formal scheme $\mathscr C$ over $\operatorname{Spf} A$ for some complete Noetherian local ring $A$ with residue field $k$ (if you are interested in mixed characteristic — which is where the term 'lift' is usually applied — then $A$ will not be a $k$-algebra).
Then the first thing you check is that such a thing is automatically smooth, since the special fibre is smooth. Or use this as your definition. Moreover, in the case of curves, the formal lift will automatically be algebraic, i.e. come from an actual lift of schemes (idea: any nonzero effective divisor on a curve is ample, so a multiple gives an embedding into projective space).
So we can just assume $C \to S = \operatorname{Spec} A$ is a smooth projective morphism, where $A$ is a complete Noetherian local ring with residue field $k$.
Now what does it mean to represent $\mathbf{Pic}^0_{C/S}$ as a scheme? The most common definitions are neatly summarised in [FGA Explained, §9.2]: you represent one of the following:
- the functor $\mathbf{Sch}_{/S} \to \mathbf{Set}$ given by $T \mapsto \operatorname{Pic}(C \times_S T)/\operatorname{Pic}(T)$;
- its Zariski sheafification;
- its étale sheafification;
- its fppf sheafification.
(Side note: from a homotopical point of view, it is probably even more natural to study the Picard stack, which is a $\mathbf G_m$-gerbe over the Picard scheme. But the gerbe structure seems to be less useful than the abelian variety structure on $\mathbf{Pic}^0$, so in many cases it is not necessary to do this.)
There are natural inclusions between these functors that are isomorphisms when $C \to S$ has a section; see [FGA Explained, Thm. 9.2.5]. By formal smoothness, such a section exists if and only if $C_0$ has a rational point (for instance when $k$ is separably algebraically closed). In general, the étale sheafification suffices, roughly because $C$ picks up a rational point after a finite separable extension.
But the important point is that in all cases, we are working on the 'big' site of all $S$-schemes, which has the advantage of giving the required functoriality for free: if any of these is representable by some scheme $\mathbf{Pic}^0_{C/S}$, then a fortiori its restriction to $\mathbf{Sch}_{/{\operatorname{Spec} k}} \subseteq \mathbf{Sch}_{/S}$ is represented by the base change $\mathbf{Pic}^0_{C/S} \times_S \operatorname{Spec} k = \mathbf{Pic}^0_{C_0/{\operatorname{Spec} k}}$.
So in summary, the required functoriality follows from representability of the Picard scheme, if stated correctly.
(I suppose this does feel like cheating, because the Jacobian has a much more elementary construction than the general formalism from FGA. But here it really helps to have the functorial point of view from the beginning!)
References.
[FGA Explained] B. Fantechi, L. Göttsche, L. Illusie, S. L. Kleiman, N. Nitsure, A. Vistoli, Fundamental algebraic geometry: Grothendieck’s FGA explained. Mathematical Surveys and Monographs 123, American Mathematical Society (2005). ZBL1085.14001.
Most chapters also available on arXiv, for instance Chapter 9 is arXiv:math/0504020.
