Given a nonsingular complete irreducible algebraic curve $C$ of genus $g$ over an algebraically closed field $k$, we can associate to it its Jacobian variety $J_C$, which is an abelian variety of dimension $g$. According to Hartshorne in Algebraic Geometry, the closed points of $J_C$ can be identified with the the group Pic$^0(C)$, the identity component of the Picard group of $C$. We have a canonical map $C \rightarrow J_C$. In almost all literature, we are only concerned with the Jacobian varieties associated to complete curves, or, as more frequently seen, projective curves (both terms are equivalent for nonsingular curves).
Then there is the notion of the generalized Jacobian $J_\mathfrak{m}$, which involves an effective divisor $\mathfrak{m}$ with support a finite set $S \subset C$, such that the canonical map $C \rightarrow J_\mathfrak{m}$ is regular outside $S$. Although the generalized Jacobian still mainly applies to complete nonsingular curves, there are instances where we can embed affine curves into their generalized Jacobian. For example, in the paper The Brauer-Manin Obstruction for Integral Points on Curves by Harari and Voloch, if $D$ is a divisor of degree $\geq 2$ of the projective line $\mathbb{P}^1_k$, then the affine curve $\mathbb{P}^1_k \backslash D$ can be embedded into its generalized Jacobian, which is an algebraic $k$-torus, where $k$ is a number field.
My question is do we have an analogous variety to associate to any affine curve?
EDIT. Perhaps I should be more specific about what I'm hoping to achieve. Based on Sasha's suggestion, I have the morphism $C \rightarrow X \rightarrow J_X$, where $X$ is the completion of $C$. We know that $J_X$ is its own dual, called the Albanese variety. Let $V$ be the $k$-torsor under $J_X$, and we have a canonical map $\phi: X \rightarrow V$. Of course, composition with the completion map gives us $\phi': C \rightarrow V$. If $V(k) \neq \emptyset$, then there is an $n$-covering $Y$ of $V$, that is, $Y$ is a $V$-torsor under $J_X[n]$.
Now we can ask, under the pull-back of $Y \rightarrow V$ via $\phi'$, do we obtain an $n$-covering of $C$ under $J_X[n]$? In other words, is $Y \times _V C$ a $C$-torsor under $J_X[n]$?