Computational fluid dynamics (CFD) problems pose significant challenges for standard reduced-order models (ROMs) based on linear subspaces, primarily due to the so-called Kolmogorov barrier. In this work, we address this limitation employing a nonlinear reduced-order modeling framework, in which a data-driven correction learns the contribution of the POD modes discarded by a very low-dimensional linear subspace approximation, effectively embedding high-frequency information into a nonlinear trial solution manifold. To mitigate the high computational cost associated with full-order model (FOM) evaluations over extensive parametric domains, we adaptively collect training samples by exploiting similarity measures based on the principal angles between pairwise parameter-dependent reduced-order subspaces. The proposed methodology is applied to the Turek–Hron CFD benchmark, which considers flow past a cylinder with a rigid flag positioned downstream. In this setting, our objective is to reduce the parametric dependence on the characteristic velocity of the parabolic inflow profile. The performance of the approach is evaluated by comparing drag and lift coefficients measured on both the cylinder and the flag against the corresponding FOM results. The nonlinear reduced-order model enables accurate predictions for parameter values outside the training set, while preserving the accuracy and interpretability of standard POD-based methods. Moreover, computational efficiency is maintained as the effective size of the reduced-order model remains that of the underlying linear ROM. Overall, the proposed framework shows strong potential for extension to coupled problems, such as fluid–structure interaction, on the same benchmark.