We define closed partially conformal vector fields and use them to give a characterization of Riemannian manifolds which admit this kind of fields as some special warped products foliated by (n-1)-umbilical hypersurfaces. In the particular case of Euclidean spaces, the foliations are then given by hyperplanes, hyperspheres or by coaxial cylinders. As an application, we obtain conditions for a constant mean curvature hypersurface to be (n-1)-umbilical.