Summary: A new superconvergence result is established for numerical solutions of elliptic problems obtained from the mixed finite element method of P. A. Raviart and J. M. Thomas [Lect. Notes Math. 606, 292–315 (1977; Zbl 0362.65089)] over rectangular partitions. The well-known optimal order error estimate in L\(^{2}\)-norm for the flux approximation is of order \({\mathcal O}(h^{k+1})\), where \(k\geq 0\) is the order of polynomials employed in the Raviart-Thomas element. The new superconvergence shows an improved accuracy of order \({\mathcal O}(h^{k+3})\) between the mixed finite element approximation and an appropriately defined local projection of the flux variable when \(k>0\). A postprocessing technique using local projection methods is proposed and analyzed in order to provide a new approximate solution with the superconvergent order \({\mathcal O}(h^{k+3})\).