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 ≃ 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}$ ≲ $ {Q}_s^2 $. Working in the limit z(1 − z)Q$^{2}$ ≪ $ {Q}_s^2 $, we predict new phenomena which would signal saturation in the SIDIS cross-section. For sufficiently low transverse momenta k$_{⊥}$ ≪ 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$_{⊥}$ ≳ Q$_{s}$, inelastic collisions take the leading role. They explore gluon saturation via multiple scattering, leading to a Gaussian distribution in k$_{⊥}$ centred around Q$_{s}$. When z(1 − z)Q$^{2}$ ≪ 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$_{⊥}$ ≫ Q$_{s}$. Still for z(1 − z)Q$^{2}$ ≪ $ {Q}_s^2 $, we also compute SIDIS cross-sections integrated over k$_{⊥}$. 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.