It is well-established that the accurate simulation of hypersonic rarefied gas flows cannot be accomplished by Navier-Stokes-Fourier (NSF) equations since they are applicable only for small deviations from local thermodynamic equilibrium. To accurately and efficiently solve this problem, a nonlinear-coupled-constitutive-relation (NCCR) model was developed from Eu's generalized hydrodynamic equations, which are consistent with the laws of irreversible thermodynamics. In this paper, Myong's decomposed solver is extended for solving three-dimensional diatomic nonlinear coupled algebraic constitutive equations. Subsequently, a reliable undecomposed hybrid algorithm is proposed for the complete solution of NCCR model through combining the merits of fixed-point and Newton's iterations. The new-developed computational method is validated by high-speed rarefied flows around a cylinder and the Apollo command module. Computation shows that the undecomposed hybrid algorithm makes a great improvement in computational robustness and accuracy. Moreover, this nonlinear constitutive model yields solutions in better agreement with DSMC than NSF model. The results indicate that NCCR model is capable of capturing the physical flow properties away from equilibrium and demonstrate its great potential capability in the further application.