With an increase in model resolution, compact high-order numerical advection scheme can improve its effectiveness and competitiveness in oceanic modeling due to its high accuracy and scalability on massive-processor computers. To provide high- quality numerical ocean simulation on overset grids, we tried a novel formulation of the fourth-order multi-moment constrained finite volume scheme to simulate continuous and discontinuous problems in the Cartesian coordinate. Utilizing some degrees of freedom over each cell and derivatives at the cell center, we obtained a two-dimensional (2D) cubic polynomial from which point values on the extended overlap can achieve fourth-order accuracy. However, this interpolation causes a lack of conservation because the flux between the regions are no longer equal; thus, a flux correction is implemented to ensure conservation. A couple of numerical experiments are presented to evaluate the numerical scheme, which confirms its approximately fourth-order accuracy in conservative transportation on overset grid. The test cases reveal that the scheme is effective to suppress numerical oscillation in discontinuous problems, which may be powerful for salinity advection computing with a sharp gradient.