In this paper, we introduce a hybrid approach that alternates between a high-fidelity model and a reduced-order model to speedup numerical simulations while maintaining accurate approximations. In particular, a residual-based error indicator is developed to determine when the reduced-order model is not sufficiently accurate and the high-fidelity model needs to be solved. Then, we propose an adaptive-extended version of the hybrid approach to update the reduced-order model with the solution snapshots generated by the high-fidelity model when the reduced-order model was not sufficiently accurate. In this way, we expect the reduced-order model to become more robust for predicting new out-of-sample solutions. The performance of the proposed method is evaluated on parametrized, time-dependent, nonlinear problems governed by the 1D Burgers' equation and 2D compressible Euler equations. The results demonstrate the accuracy and computational efficiency of the adaptive hybrid approach with respect to the high-fidelity model.