Lattice stress arises in thin bilayers and superlattices owing to the mismatch between the lattices of two different materials. In this paper, the dependence of the (CdCo)-Cd-111 magnetic hyperfine field in Re/Co thin bilayers and superlattices on the thickness of the layers is measured with the Perturbed Angular Correlations technique, and explained by the strain induced in the Co layers. The stress is relieved through the creation of stacking fault defects in the Co structure. It is larger in the bilayers than in the superlattices, where a worse-defined structure leads to additional stacking faults, and hence to a faster relief of the stress.