A new family of finite element methods, named Multiscale Hybrid-Mixed method (or MHM for short), aims to solve reactive-advective dominated problems with multiscale coefficients on coarse meshes. The underlying upscaling procedure transfers to the basis functions the responsibility of achieving high orders of accuracy. The upscaling is built inside the general framework of hybridization, in which the continuity of the solution is relaxed a priori and imposed weakly through the action of Lagrange multipliers. This characterizes the unknowns as the solutions of local problems with Robin boundary conditions driven by the multipliers. Such local problems are independent of one another, yielding a process naturally shaped for parallelization and adaptivity. Moreover, the multiscale decomposition indicates a new adaptive algorithm to set up local spaces defined using a face-based a posteriori error estimator. Interestingly, it also embeds a postprocessing of the dual variable (flux) which preserves local conservation properties of the exact solution. Extensive numerical validations assess the claimed optimal rates of convergence, the robustness of the method with respect to the model's coefficients, and the adaptivity algorithm.