It can be found, at Hartshorne exercise 4.2.7 for example, that in the case where $char(k) \neq 2$ we have a nice correspondence between etale degree 2 covers of a curve $C$ and 2-torsion in $JC$. This correspondence, as constructed in Hartshorne, breaks down in the case where $char(k) = 2$.

However it has been suggested in https://arxiv.org/pdf/1004.2267.pdf , at the bottom of section 4.7, that "remarks following proposition 4.13 in chapter 3 of Milne's Etale cohomology" show that some correspondence between $JC[2](k)$ and etale degree 2 covers of $C$ still exists. Does anyone know how to construct such a correspondence explicitly? I believe it is erroneous (there is no proposition 4.13, it is a lemma!)

Otherwise, do etale degree 2 covers of $C$ correspond to anything else? Artin-Schreier theory suggests that it should be $H^1(C,\mathcal{O}_C)^F$. However from such a nontrivial cover $\pi:X \to C$ we also obtain a nontrivial extension of $\mathcal{O}_C$-algebras $$0 \to \mathcal{O}_C \to \pi_*\mathcal{O}_X \to \mathcal{O}_C \to 0$$ by looking at affines of $C$ and writing explicit Artin-Schreier extensions, and observing that the cocycles on $\pi_*\mathcal{O}_X$ are unipotent $2\times2$ matrices. Is there any way to relate this to $JC$?