Convected particle domain interpolation (CPDI) is a recently developed technique for more accurately approximating material point method (MPM) integrals by replacing the shape functions with interpolations of the shape functions to corners of particle domains that are tracked as parallelograms in 2-D (or parallelepipeds in 3-D). In this paper, the CPDI method is enhanced to (1) more accurately track particle domains with very little computational overhead in comparison to the original CPDI, (2) remove gaps/overlapping between particle domains, and (3) give more flexibility in choosing particle domain shape in the initial configuration. This enhanced CPDI method is then enriched to accurately solve weak discontinuities in the displacement field across a material interface that passes through the interior of a grid cell. The new enriched CPDI2 method is demonstrated using one- and two-dimensional examples.