This question has been answered in a MathOverflow cross-post.
Suppose $\mathcal C$ is a category and $S \subseteq \operatorname{Mor}(\mathcal C)$ is some collection of morphisms in $\mathcal C$. Let $\operatorname{Loc}(S) $ be the full subcategory of $\mathcal C$ on the $S$-local objects. Then we have $$ \operatorname{Loc}(S) \stackrel{\iota}{\hookrightarrow} \mathcal C \stackrel{P}{\twoheadrightarrow} \mathcal C[S^{-1}], $$ where $\iota$ is the inclusion and $P$ is the canonical functor to the localization of $\mathcal C$ by $S$. Then if $P$ admits a fully faithful right adjoint $r : \mathcal C[S^{-1}] \to \mathcal C$ (actually, I think any right adjoint to $P$ will be fully faithful?), then $r$ will be equivalent to $\iota$ (as objects of $\mathrm{Cat}/\mathcal{C}$), and so furthermore $\iota$ will have a left adjoint which is equivalent to $P$ (as objects of $\mathcal C/\mathrm{Cat}$).
I'd like to know about the converse situation: What if $\iota$ admits a left adjoint $L : \mathcal C \to \operatorname{Loc}(S)$? Does it follow that $L$ will be equivalent to $P$ (and hence that $P$ will have a right adjoint which is equivalent to $\iota$)?
I suspect the answer is "not necessarily", because the nLab page on reflective localizations lists two facts which would clearly be implied by the converse I want but which do not clearly imply it. In particular, it says that $L$ will be a localization of $\mathcal C$ by the class of morphisms $L$ inverts—but this is in general a strict superclass of $S$, so this fact alone would not imply $L$'s equivalence to $P$. It also says that $L$ will have a universal property similar to the one for a localization by $S$, but which only applies to left adjoint functors that invert $S$.