The authors consider the elliptic problem: \(-\nabla \cdot \left( \mathcal{ K\nabla }u\right) =f\) in \(\Omega\), \(\mathcal{K}\nabla u\cdot \mathbf{n}=0\) on \(\partial \Omega _{N}\), \(u=g_{D}\) on \(\partial \Omega _{D}\), where some regularity conditions are imposed and \(\mathcal{K=}\left\{ \mathcal{K} _{ij}\right\} \in \left[ L^{\infty }\left( \Omega \right) \right] ^{d\times d}\) is a uniformly elliptic symmetric tensor. Specially designed to incorporate multiple scales into the construction of basis functions, the proposed multiscale hybrid-mixed finite element method (MHM) relaxes the continuity of the primal variable through the action of Lagrange multiplies but also assures the strong continuity of the dual variable. The authors prove the existence and uniqueness of the solution for the MHM, show that the method is well posed, provide a best approximation, establish a priori error estimates showing optimal convergence in natural norms and give a face-based a posteriori estimator. The MHM may be used in parallel computing environments in order to solve multiscale boundary value problems with precision on coarse meshes.