I think using motivic cohomology, it ca can be shown that $\mathrm{Pic}(X \times Y) = \mathrm{Pic}(X) \oplus \mathrm{Hom}(\mathrm{Jac}(X), \mathrm{Jac}(Y)) \oplus \mathrm{Pic}(Y)$, or something like that., or perhaps $\mathrm{NS}$ instead of $\mathrm{Pic}$.
I think using motivic cohomology, it ca be shown that $\mathrm{Pic}(X \times Y) = \mathrm{Pic}(X) \oplus \mathrm{Hom}(\mathrm{Jac}(X), \mathrm{Jac}(Y)) \oplus \mathrm{Pic}(Y)$, or something like that.