New higher-order accurate spatial interpolation schemes for the finite-volume method (FVM) on unstructured Cartesian grids are proposed. In the FVM formulation, the accuracy of both the interpolation of the variable and the flux quadrature affects the formal spatial accuracy. In this paper, the accuracy of the interpolation using the gradients of the variable is considered. The gradient is conventionally approximated using Green-Gauss method or weighted least-square method, but the accuracy is less than second-order; therefore, the gradient is reformulated using the gradient values saved in the adjacent cells. This corresponds to virtually extending the stencils, and fifth-order accuracy is achieved with referring to only the adjacent cells. This formulation is based on the advantage of Cartesian grids, i.e. the most part of the grid is uniform. In the twodimensional problem, the present schemes show better preservability of an isentropic vortex. Finally, a two-dimensional direct acoustic simulation is conducted. The result shows that the present schemes have better wave resolution, and is consistent with the result using conventional third-order scheme on finer grids. It is also confirmed that the present scheme has enough stability in the problem. The present schemes have good balance between the resolution and the computational cost compared with the conventional higher-order schemes for unstructured grids.