We propose a fully conservative high-order upwind multi-moment method for the conservation equation. The proposed method is based on a third-order polynomial interpolation function and semi-Lagrangian formulation and is a variant of the constrained interpolation profile conservative semi-Lagrangian scheme with third-order polynomial function method. The third-order interpolation function is constructed based on three constraints in the upwind cell (two boundary values and a cell average) and a constraint in the downwind cell (a cell center value). The proposed method shows fourth-order accuracy in a benchmark problem (sine wave propagation). We also propose a less oscillatory formulation of the proposed method. The less oscillatory formulation can minimize numerical oscillations. These methods were validated through scalar transport problems, and compressible flow problems (shock tube and 2D explosion problems). Copyright (C) 2016 John Wiley & Sons, Ltd.