We examine the possibility of introducing a new class of so called non-iterative corrections to coupled cluster energies, based on the connected moments expansion (CMX). Approximate Coupled Cluster (CC) wave functions are used as reference states and then the "improved energies" can be formally obtained either by CMX in terms of the moments of CC similarity transformed Hamiltonian, or in CMX of the usual Hamiltonian. Numerical results are given for some model systems that show the superiority of the latter approach.