The material point method (MPM) is an effective computational tool for simulating problems involving large deformations. However, its direct mapping of the material-point data to the background grid frequently leads to severe inaccuracies. The standard function reconstruction techniques can considerably decrease these errors, but do not always guarantee the conservation of the total mass and linear momentum as the MPM algorithm does. In this paper, we introduce a novel technique, called Taylor Least Squares (TLS), which combines the Least Squares approximation with Taylor basis functions to reconstruct functions from scattered data. Within each element, the TLS technique approximates quantities of interest, such as stress and density, and when used with a suitable quadrature rule, it conserves the total mass and linear momentum after mapping the material-point information to the grid. The numerical and physical properties of the reconstruction technique are first illustrated on one- and two-dimensional functions. Then the TLS technique is tested as part of MPM, Dual Domain Material Point Method (DDMPM), and B-spline MPM (BSMPM) on a one-dimensional problem experiencing small and large deformations. The obtained results show that applying the TLS approximation significantly improves the accuracy of the considered versions of the material point method, while preserving the physical properties of the standard MPM.