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, an error indicator based on a reduced-order approximation of the error 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 new solution snapshots. In particular, an interesting application concerns the acceleration of the offline stage of ROMs. The performance of the proposed method is finally 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.