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Two variants of the MCV3 scheme are presented based on a flux reconstructionformulation. Different from the original multi-moment constrained finite volumemethod of third order (MCV3), the multi-moment constraints are imposed at thecell center on the point value, the first and second order derivatives. Thecontinuity of the flux function at cell interfaces are also used as theconstraints to ensure the numerical conservation. Compared to the original MCV3scheme, both two variants have higher numerical accuracy and less restrictiveCFL condition for computational stability. Moreover, without the need to solvederivative Riemann problem at cell boundaries, the new schemes benefit theimplementations in arbitrary quadrilateral in 2D and hexahedron in 3D.