In this study, we demonstrate an acceleration of flexible generalized minimal residual algorithm (FGMRES) implemented with the method of moments and the fast multipole method (FMM), based on a combined tangential formulation. For the implementation of the FGMRES incorporated with the FMM concept, we propose a new definition of the truncation number for the FMM operator within the inner solver. The proposed truncation number provides an optimal variable preconditioner by controlling the accuracy and computational cost of the inner iteration. Moreover, to further accelerate the convergence, we introduce the concept of a multistage preconditioner. Numerical experiments reveal that our new version of FGMRES, based on the proposed truncation number for the inner solver and the multistage preconditioner, achieves outstanding acceleration of the convergence for large-scale and practical electromagnetic scattering and radiation problems with several levels of geometrical complexity.