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