The multilayer model proposed in this paper is a generalization of the multilayer non-hydrostatic model for shallow granular flows [28], the multilayer model with µ(I) rheology [25], and the monolayer model with weakly non-hydrostatic pressure for dry granular flows [29]. We show that the proposed model verifies a dissipative energy balance. A well-balanced numerical scheme is proposed to solve the equations based on a projection method and a hydrostatic reconstruction for the Coulomb friction terms. In order to reduce the computational cost associated with solving the linear system of the projection method, a precomputing of the initial guess for an iterative solver is proposed. This strategy allows us to reduce the computational time by around 70% when 20 layers are considered. In the numerical tests, we show that the proposed model can recover the in-depth velocity profiles typically observed in lab experiments and capture the flow/no-flow interface that appears in granular avalanches. During the initial stage of granular collapse simulations, the model is shown to improve the approximation of the mass profiles compared to other models and to predict the parabolic shape of the front velocity evolution with time, as observed in lab experiments. Interestingly, our numerical tests show that the ability of the granular flow to overcome obstacles strongly depends on the model used, which is of strong interest for landslide hazard assessment.