For this question I will let the overly ambiguous word curve mean: smooth projective and connected curve over $\mathbb C$ (or equivalently a smooth compact Riemann-Surface).

Let $C$ be a curve over $\mathbb C$. And suppose that $f: D \to C$ and $g:D \to E$ are dominant maps of curves. Then the pair $(f,g)$ forms a correspondence from $C$ to $E$, and one gets a map between Jacobians $f_* \circ g^* : J(E) \to J(C)$. Then the image of $J(C)$ under this correspondence is going to be a sub abelian variety of $J(C)$.

My questions is, is it possible to get all simple sub abelian varieties of $J(C)$ from correspondences in this way?