Local coupled cluster theory is based on (i) a restriction of the list of pairs (or triples) of occupied molecular orbitals, and (ii) a truncation of the virtual space to orbital pair (or triple) specific subspaces. The latter is motivated by an exponential decay of the contributions to the pair energy with respect to the distance between related local occupied and virtual orbitals; the former only by a polynomial R(-6) decay with respect to the distance R between the two occupied orbitals of the pair. Consequently, the restriction of the pair list is more critical, and contributions of pairs should not be neglected unless the corresponding interorbital distance is really large. In local coupled cluster theory pairs are usually discriminated on the basis of the interorbital distance, or the size of the 2nd order Møller-Plesset perturbation theory (MP2) estimate of the pair energy. Only strong pairs are treated at the full coupled cluster level, while weak pairs are treated just at the level of MP2. Yet MP2 might be problematic in certain cases, for example, π-stacking is badly described by MP2, etc. We propose to substitute the MP2 treatment of weak pairs by an approach based on ring-CCD by including third-order diagrams with R(-6) decay behavior. Such an approach is clearly superior; it provides higher accuracy, while the computational cost is not significantly higher than that of a MP2 based treatment of weak pairs.