Here we develop the Gauss-Legendre-spherical-t (GLST) algorithm that overcomes the aforementioned issues of SGT. It truncates the integral representation of the long-range term to a finite range while keeping the resulting approximation accurate within the target accuracy. Conversion of the integral to a finite range allows using the Gauss-Legendre quadrature . Unlike the Gauss-Hermite quadrature, most of the Gauss-Legendre quadrature nodes contribute meaningfully to the approximation without having to deal with negligible tails of the integral suppressed by the Gaussian weight. This change significantly reduces the number of cubature nodes to be used. Instead of a treecode approach, we divide the simulation box into cells where the number does not need to be in powers of 2. By taking advantage of the fact that smaller number of cubature nodes are needed for larger distances, we group faraway cells and assign cubatures with smaller sizes.