  Let $X$ be a smooth projective complex variety of maximal Albanese dimension,
and let $L \to X$ be a big line bundle. We prove that the moving Seshadri
constants of the pull-backs of $L$ to suitable finite abelian \'etale covers of
$X$ are arbitrarily large. As an application, given any integer $k\geq 1$,
there exists an abelian \'etale cover $p\colon X' \to X$ such that the adjoint
system $\big|K_{X'} + p^*L \big|$ separates $k$-jets away from the augmented
base locus of $p^*L$, and the exceptional locus of the pull-back of the
Albanese map of $X$ under $p$.
