By a theorem of Matsusaka, every abelian variety $A$ over an algebraic closed field $k$ is a *quotient* of a Jacobian. Now just apply Matsusaka's theorem to $A^{\vee}$, and dualize. I think that the slightly weaker version where the field is infinite is proved somewhere in Milne's lecture notes on Abelian Varieties. **Reference.** T. Matsusaka: On a generating curve of an Abelian variety, *Nat. Sci. Rep. Ochanomizu Univ.* **3** 1-4, (1952). Quoting from P. Samuel review on MathSciNet: >An abelian variety $A$ is said to be generated by a variety $V$ (and a mapping $f$ of $V$ into $A$) if $A$ is the group generated by $f(V)$. It is proved that every abelian variety $A$ may be generated by a curve defined over the algebraic closure of $\mathrm{def}(A)$.