Thin-walled composite structures, valued for their lightweight potential, find extensive application in the aerospace and shipbuilding industries. However, the stability behavior of these structures needs to be considered. As a locally post-buckled structure demonstrates the capacity to withstand increasing loads without immediate failure, it necessitates not only a buckling analysis but also a post-buckling analysis to fully leverage its lightweight potential. In the post-buckling regime, mode changes represent a change in buckling deformation after a critical load is reached, associated with a loss of stiffness. These changes can lead to failure of the structure. Therefore, it is important to analyze them in the context of a post-buckling analysis. However, due to the highly non-linear behavior, bifurcation and limit points of these problems are difficult to solve. In addition, not every solution to the equilibrium equations is a stable or physically relevant solution. Therefore, exploring and classifying the solutions in the post-buckling regime is an additional challenge. Based on the Riks method a procedure for tracing those equilibrium paths in an efficient yet accurate way is developed and results regarding the plate buckling of rectangular composite plates under uniaxial compression accounting for bending-twisting coupling are presented. This research contributes to the development of computationally efficient Ritz-methods for designing optimized thin-walled composite structures. This is achieved by reducing the number of integrals to be evaluated and simultaneously efficiently tracing the equilibrium path in the post-buckling regime.