We have recently introduced the tensor hypercontraction (THC) method for electronic structure, including MP2. Here, we present an algorithm for THC-MP2 that lowers the memory requirements as well as the prefactor while maintaining the formal quartic scaling that we demonstrated previously. We also describe a procedure to optimize quadrature grids used in grid-based least-squares (LS) THC-MP2. We apply this algorithm to generate grids for first-row atoms with less than 100 points/atom while incurring negligible errors in the computed energies. We benchmark the LS-THC-MP2 method using optimized grids for a wide variety of tests sets including conformational energies and reaction barriers in both the cc-pVDZ and cc-pVTZ basis sets. These tests demonstrate that the THC methodology is not limited to small basis sets and that it incurs negligible errors in both absolute and relative energies.