The (n+1)-sphere contains a simple family of constant mean curvature (CMC) hypersurfaces equal to products of a p-sphere and a 1-sphere of different radii, called the generalized Clifford hypersurfaces. This paper demonstrates that two new, topologically non-trivial CMC hypersurfaces resembling a pair of neighbouring generalized Clifford tori connected to each other by small catenoidal bridges at a sufficiently symmetric configuration of points can be constructed by perturbative PDE methods. That is, one can create an approximate solution by gluing a rescaled catenoid into the neighbourhood of each point; and then one can show that a perturbation of this approximate hypersurface exists, which satisfies the CMC condition. The results of this paper generalize previous results of the authors.