An enhanced implementation of the random ray method (TRRM), a stochastic adaptation of the method of characteristics, is presented. This implementation generalizes from a traditional flat source (FS) approximation to a linear source (LS) approximation, and from the standard isotropic scattering approximation to an anisotropic approximation, up to P3. On the two-dimensional C5G7 benchmark, LS alone enables a 1.4× and 1.7× enhancement in run time for parallel and serial executions, respectively, without compromising accuracy. Further testing on the three-dimensional C5G7 Rodded B benchmark demonstrates that LS enables significant axial coarsening and reduction in ray populations. This results in a 5.3× speedup in run time while still offering comparable accuracy. On the Babcock and Wilcox 1484 Core II benchmark, incorporating anisotropic scattering in TRRM up to P3 with a FS decreased keff errors (~110 pcm) and maximum and average pin power errors. LS with anisotropic scattering (LSPN), showed similarly improved keff errors while benefiting from a coarser mesh. The linear isotropic flat anisotropic (LIFA) method, relative to LSPN, offers further run time and memory savings of 4.1× and 1.6×, respectively, for P3, while maintaining comparable accuracy to anisotropic FS on more refined meshes.