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A number of different multiscale methods have been developed as a robust alternative to upscaling and as a means for accelerated reservoir simulation of high-resolution geomodels. In their basic setup, multiscale methods use a restriction operator to construct a reduced system of flow equations on a coarser grid, and a prolongation operator to map pressure unknowns from the coarse grid back to the original simulation grid. The prolongation operator consists of basis functions computed numerically by solving localized flow problems. One can use the resulting multiscale solver both as a CPR-preconditioner in fully implicit simulators or as an efficient approximate iterative linear solver in a sequential setting. The latter approach has been successful implemented in a commercial simulator. Recently, we have shown that you can obtain significantly faster convergence if you instead of using a single pair of prolongation-restriction operators apply a sequence of such operators, where some of the operators adapt to faults, fractures, facies, or other geobodies. Herein, we present how you can accelerate the convergence even further, if you also include additional basis functions that capture local changes in the pressure.