A D2Q9 Hybrid Lattice Boltzmann Method (HLBM) is proposed for the simulation of both compressible subsonic and supersonic flows. This HLBM is an extension of the model of Feng et al. [1], which has been found, via different test cases, to be unstable for supersonic regimes. To circumvent this limitation, we propose:: (1) a new discretization of the lattice closure correction term that makes possible the simulation of supersonic flows, (2) a corrected viscous stress tensor that takes into account polyatomic gases, and (3) a novel discretization of the viscous heat production term fitting with the regularized formalism. The result is a hybrid method that resolves the mass and momentum equations with an LBM algorithm, and resolves the entropy-based energy equation with a finite volume method. This approach fully recovers the physics of the Navier-Stokes-Fourier equations with the ideal gas equation of state, and is valid from subsonic to supersonic regimes. It is then successfully assessed with both smooth flows and flows involving shocks. The proposed model is shown to be an efficient, accurate, and robust alternative to classic Navier-Stokes methods for the simulation of compressible flows.