A low-cost approach for stochastically sampling static exchange during TDHF-type propagation is presented. This enables the use of an excellent hybrid DFT starting point for stochastic GW quasiparticle energy calculations. Generalized Kohn-Sham molecular orbitals and energies, rather than those of a local-DFT calculation, are used for building the Green's function and effective Coulomb interaction. The use of an optimally tuned hybrid diminishes the starting point dependency in one-shot stochastic GW, effectively avoiding the need for self-consistent GW iterations.