  Using the dipole picture for electron-nucleus deep inelastic scattering at
small Bjorken $x$, we study the effects of gluon saturation in the nuclear
target on the cross-section for SIDIS (single inclusive hadron, or jet,
production). We argue that the sensitivity of this process to gluon saturation
can be enhanced by tagging on a hadron (or jet) which carries a large fraction
$z \simeq 1$ of the longitudinal momentum of the virtual photon. This opens the
possibility to study gluon saturation in relatively hard processes, where the
virtuality $Q^2$ is (much) larger than the target saturation momentum $Q_s^2$,
but such that $z(1-z)Q^2\lesssim Q_s^2$. Working in the limit $z(1-z)Q^2\ll
Q_s^2$, we predict new phenomena which would signal saturation in the SIDIS
cross-section. For sufficiently low transverse momenta $k_\perp\ll Q_s$ of the
produced particle, the dominant contribution comes from elastic scattering in
the black disk limit, which exposes the unintegrated quark distribution in the
virtual photon. For larger momenta $k_\perp\gtrsim Q_s$, inelastic collisions
take the leading role. They explore gluon saturation via multiple scattering,
leading to a Gaussian distribution in $k_\perp$ centred around $Q_s$. When
$z(1-z)Q^2\ll Q^2$, this results in a Cronin peak in the nuclear modification
factor (the $R_{pA}$ ratio) at moderate values of $x$. With decreasing $x$,
this peak is washed out by the high-energy evolution and replaced by nuclear
suppression ($R_{pA}<1$) up to large momenta $k_\perp\gg Q_s$. Still for
$z(1-z)Q^2\ll Q_s^2$, we also compute SIDIS cross-sections integrated over
$k_\perp$. We find that both elastic and inelastic scattering are controlled by
the black disk limit, so they yield similar contributions, of zeroth order in
the QCD coupling.
