The GW approximation to the electron self energy has become a standard method for ab initio calculation of excited-state properties of condensed-matter systems. In many calculations, the GW self energy operator, Σ, is taken to be diagonal in the density functional theory (DFT) Kohn-Sham basis within the G$_0$W$_0$ scheme. However, there are known situations, where this diagonal G$_0$W$_0$ approximation starting from DFT is inadequate. We present two schemes to resolve such problems. The first, we called sc-COHSEX+GW, involves construction of an improved mean-field (MF) using the static limit of GW, known as COHSEX (Coulomb hole and screened exchange) which is significantly simpler to treat than GW. In this scheme, Σ(ω) is constructed and taken to be diagonal in the COHSEX orbitals after the system is solved self-consistently within this formalism. The second method is called off diagonal-COHSEX GW (od-COHSEX+GW). In this method, one does not self-consistently change the mean-field starting point but diagonalizes the COHSEX Hamiltonian within the Kohn-Sham basis to obtain quasiparticle wave functions and uses the resulting orbitals to construct the GW Σ in the diagonal form. We apply both methods to a molecular system, silane, and to two bulk systems, Si and Ge under pressure. For silane both methods give good quasiparticle wave function and energy. Both methods give good band gaps for bulk silicon and maintain good agreement with experiment. Further, the sc-COHSEX+GW method solves the qualitatively incorrect DFT mean-field starting point (having a band overlap) in bulk Ge under pressure.