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 $z(1-z)Q^2lesssim Q_s^2$. Working in the limit $z(1-z)Q^2ll Q_s^2$, we predict new phenomena which would signal saturation in the SIDIS cross-section. For sufficiently low transverse momenta $k_perpll Q_s$ 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 $k_perpgtrsim Q_s$, 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 $z(1-z)Q^2ll Q^2$, 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 $k_perpgg Q_s$. Still for $z(1-z)Q^2ll Q_s^2$, 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.
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