The Multiscale Hybrid-Mixed (MHM) method is a multiscale finite element method based on a hybrid weak formulation, originally proposed for problems linked to flow in porous media. Its starting point is a variational formulation that guarantees that the flux variable is H(div, Ω)-conforming, but this property is lost when the local problems, in their elliptic form, are discretized using standard finite element methods. In this work, we close that gap by proposing and analyzing a new flux reconstruction, computed element-wise, that belongs to H(div, Ω) for the MHM method. This reconstruction converges optimally in the L^2( Ω)-norm, and its divergence is the projection onto the finite element space of the right-hand side datum of the problem, and thus it is super-convergent. As a by-product of the reconstruction technique, a fully computable \textit{a posteriori} error estimator is presented and analyzed. These theoretical results are validated experimentally via numerical computations.