Aeroacoustic simulations are extremely sensitive to numerical errors. To achieve a high-order of accuracy, optimized finite difference schemes, such as the well-known dispersion-relation-preservation schemes, and multi-block structured meshes are often used. Challenges appear in communicating the flow information between neighboring blocks, and sophisticated interface treatments are required. The characteristic interface condition has shown to be successful in applications in previous publications, and yet limitations exist that are identified in the present study. This paper proposes a new interface treatment based on a unified finite difference and finite volume discretization. Applications to several benchmark cases show an encouraging improvement as compared to the characteristic interface treatments.