In low dimensions, we prove that every complete finite-index CMC hypersurface is either minimal or compact, provided that the ambient manifold is a Riemannian product of a closed manifold with non-negative sectional curvature and a Euclidean factor. This is related to a question posed by do Carmo and extends previously known results. Our proof builds on the recent resolution of the stable Bernstein problem in R^6. We show that a complete finite-index CMC hypersurface immersed in low-dimensional hyperbolic space with mean curvature |H|>7 is compact and present new examples of stable CMC hypersurfaces. These results give a partial answer to a question posed by Chodosh in his survey for the ICM.