Standard Full-waveform inversion (FWI) often suffers from poor sensitivity to deep features of the subsurface model. To alleviate this problem, we propose a hybrid linear and non-linear optimization method to enhance the FWI results. In this method, iterative least-squares reverse-time migration (LSRTM) is used to estimate the model update at each nonlinear iteration, and the number of LSRTM iterations is progressively increased after each non-linear iteration. With this method, model updating along deep reflection wavepaths are automatically enhanced, which in turn improves imaging below the reach of diving-waves. This hybrid linear and non-linear FWI algorithm is implemented in the space-time domain to simultaneously invert the data over a range of frequencies. A multiscale approach is used where higher frequencies are iteratively incorporated into the inversion. Synthetic data are used to test the effectiveness of reconstructing both the high- and low-wavenumber features in the model without relying on diving waves in the inversion. We apply the method to Gulf of Mexico field data and illustrate the improvements after several iterations. Results show a significantly improved migration image in both the shallow and deep sections.