This work seeks to improve upon an existing formulation of the Method of Characteristics (MOC) with a Linear Source Approximation (LSA) for problems that use nonconstant cross sections like multiphysics feedback and the two-dimensional/one-dimensional (2D/1D) formulation. The previous LSA formulation for lattice physics calculations uses precomputed coefficients that are dependent on the multigroup total or transport cross sections, and the method can be demonstrated to lack robustness when there are negative sources. In this paper, the method is reformulated to eliminate the cross-section dependence of the precomputed coefficients without adding additional operations, and a more robust formulation is also developed to prevent the calculation of negative sources. Thus, the method has increased efficiency and robustness in multiphysics and 2D/1D simulations. The new method is implemented in the MPACT code and tested on several light water reactor problems. The numerical results show that with the new Linear Source formulation, the number of mesh elements can be significantly reduced while maintaining accuracy, resulting in reduced run time and memory usage. Furthermore, our results demonstrate improved efficiency for cases with depletion, thermal-hydraulic feedback, and in three-dimensional (2D/1D) calculations without any robustness issues.