Full-waveform inversion (FWI) attempts to resolve an ill-posed non-linear optimization problem in order to retrieve the unknown subsurface model parameters from seismic data. Generally, FWI fails to obtain an adequate representation of models with large high-velocity structures over a wide region, like salt bodies and the sediments beneath them, in the absence of low frequencies in the recorded seismic signal, due to non-linearity and non-uniqueness. We alleviate the ill-posedness of FWI associated with datasets affected by salt bodies using model regularization. We split the optimization problem into two parts: first, we minimize the data misfit and the total variation in the model, seeking to achieve an inverted model with sharp interfaces; and second, we minimize sharp velocity drops with depth in the model. Unlike conventional industrial salt flooding, our proposed technique requires minimal human intervention and no information about the top of the salt. Those features are demonstrated on datasets of the BP 2004 and Sigsbee2A models, synthesized from a Ricker wavelet of dominant frequency 5.5 Hz and minimum frequency 3 Hz. We initiate the inversion process with a simple model in which the velocity increases linearly with depth. The model is well retrieved when the same constant density acoustic code is used to simulate the observed data, which is still one of the most common FWI tests. Moreover, our technique allows us to reconstruct a reasonable depiction of the salt structure from the data synthesized independently with the BP 2004 model with variable density. In the Sigsbee2A model, we manage even to capture some of the fine layering beneath the salt. In addition, we show the versatility of our method on a field dataset from the Gulf of Mexico.