We present a thermal device based on the near-field interaction between two substrates made of a polar and a metal-insulator-transition material. As a result of the temperature dependence of the optical properties, this device acts as a thermal rectifier, implying a strong asymmetry in the heat flux when reversing the two temperatures. By covering both substrates with a graphene sheet we show a significant enhancement of rectification coefficient. By investigating the flux spectral properties along with its distance dependence, we prove that this enhancement is associated to a change in the power-law dependence of heat flux with respect to the separation distance in the electrostatic regime due to the presence of graphene sheets. Our results highlight the promising role of graphene-based hybrid structures in the domain of nanoscale thermal management.