The reaction path connects a chemical potential energy landscape and the conceptual descriptions of chemical mechanisms and reactivity. In recent years, a class of predictor-corrector integrators has been developed and shown to provide an excellent compromise between computational efficiency and numerical accuracy. Models based on projected frequencies along the reaction path and coupling matrix elements, such as Reaction Path Hamiltonian (RPH) and Unified Reaction Valley Approach (URVA), require highly accurate integration of the reaction path. In this report, the Euler Predictor-Corrector (EulerPC) and Hessian-based Predictor-Corrector (HPC) methods are shown to be inadequate for studying reaction path curvature, which is a central component of the RPH and URVA models. The source of this apparent failure is explored, and a solution is developed. Importantly, the resulting enhanced EulerPC and HPC integrators do not require more intensive CPU or memory requirements than their predecessors.