  In this paper, we introduce a class of super Adler-type operators associated
with the Lie superalgebra $\mathfrak{gl}(m|n)$. We show that these operators
generate Poisson vertex superalgebras which are isomorphic to the classical
$\mathcal{W}$-superalgebras associated with $\mathfrak{gl}(m|n)$ and some
rectangular nilpotent elements. We use this isomorphism to construct integrable
hierarchies on these rectangular $\mathcal{W}$-superalgebras.
