The linearized constitutive relations in the Navier-Stokes-Fourier equations are only applicable to near-equilibrium flows. Nonlinear coupled constitutive relations (NCCR) are emerging as a solution for extremely non-equilibrium flows, such as hypersonic rarefied gas flows. However, due to its highly nonlinear and strongly coupled nature, solving the NCCR equations is a key challenge. To address this key challenge, a hybrid iterative algorithm is proposed by combining the advantages of Newton’s iteration method and fixed-point iteration method. In order to demonstrate the effectiveness of hybrid iterative strategy, we investigate the NCCR equations from the perspective of topology. Topology has long been a fascinating topic in the study of the fluid mechanics. Topological results for the NCCR equations show that neither the fixed-point method (especially in the expansion region) nor the Newton’s iteration method (especially in the compression region) are satisfactory for convergence. However, the hybrid iterative strategy can construct smooth NCCR topologies in both expansion and compression regions. This work provides guidance for solving NCCR equations in a stable and efficient manner.