Full-waveform inversion (FWI), proposed by Lailly and Tarantola in the 1980s, is considered to be the most promising data-driven tool for automatically building velocity models. Many successful examples have been reported using FWI to update shallow sediments, gas pockets, and mud volcanoes. However, successful applications of FWI to update salt structures had almost only been seen on synthetic data until recent progress at the Atlantis Field in the Gulf of Mexico. We revisited some aspects of FWI algorithms to minimize cycle-skipping and amplitude discrepancy issues and derived an FWI algorithm that is able to build complex salt velocity models. We applied this algorithm to a variety of data sets, including wide-azimuth and full-azimuth (FAZ) streamer data as well as ocean-bottom-node data, with different geologic settings in order to demonstrate the effectiveness of the method for salt velocity updates and to examine some fundamentals of the salt problem. We observed that, in multiple cases, salt velocity models from this FWI algorithm produced subsalt images of superior quality. We demonstrate with one FAZ streamer data example in Keathley Canyon that we do not necessarily need very high frequency in FWI for subsalt imaging purposes. Based on this observation, we envision that sparse node for velocity acquisition may provide appropriate data to handle large and complex salt bodies with FWI. We believe the combination of advanced FWI algorithms and appropriate data acquisition will bring a step change to subsalt imaging.