In this paper, we extend the finite volume direct average micromechanics to enable the use of quadrilateral subcells. To do this work, the quadrilateral subcells are used to discretize the repeating unit cells first. Then the average displacement and traction defined on the boundary of the subcell are evaluated by direct integral method. This contrasts with the original formulation in which all of the subcells are rectangular. Following the discretization, the cell problem is defined by combining the directly volume-average of the subcell stress equilibrium equations with the displacement and traction continuity in a surface-average sense across the adjacent subcell faces. In order to assemble the above equations and conditions into a global equation system, the global and local number systems, which index the boundary of subcell in different manners, are employed by the extended method. Finally, the global equation system is solved and the solutions give the formulations of the microstress field and the global elastic moduli of material. The introduction of quadrilateral subcells increases the efficiency of modeling the material’s microstructure and eliminates the stress concentrations at the curvilinear bimaterial corners. Herein, the advantage of the extension is presented by comparing the global moduli and local stress fields predicted by the present method with the corresponding results obtained from the original version.