This work focuses on the dynamics around a partially elliptic, lower dimensional torus of a real analytic Hamiltonian system. More concretely, we investigate the abundance of invariant tori in the directions of the phase space corresponding to elliptic modes of the torus. Under suitable (but generic) non-degeneracy and non-resonance conditions, we show that there exists plenty of invariant tori in these elliptic directions, and that these tori are organized in manifolds that can be parametrized on suitable Cantor sets. These manifolds can be seen as ``Cantor centre manifolds'', obtained as the nonlinear continuation of any combination of elliptic linear modes of the torus. Moreover, for each family, the density of the complementary of the set filled up by these tori is exponentially small with respect to the distance to the initial torus. These results are valid in the limit cases when the initial torus is an equilibrium point or a maximal dimensional torus. It is remarkable that, in the case in which the initial torus is totally elliptic, we can derive Nekhoroshev-like estimates for the diffusion time around the torus. Due to the use of weaker non-resonance conditions, these results are an improvement on previous results by the authors.