When present in the subsurface salt bodies impact the complexity of wave-equation-based seismic imaging techniques, such as least-squares reverse-time migration, and full-waveform inversion (FWI). Typically, the Born approximation used in every iteration of least-squares-based inversions is incapable of handling the sharp, high-contrast boundaries of salt bodies. We develop a variance-based method for reconstruction of velocity models to resolve the imaging and inversion issues caused by salt bodies. Our main idea lies in retrieving useful information from independent updates corresponding to FWI at different frequencies. After several FWI iterations we compare the model updates by considering the variance distribution between them to identify locations most prone to cycle skipping. We interpolate velocities from the surrounding environment into these high-variance areas. This approach allows the model to gradually improve from identifying easily resolvable areas and extrapolating the model updates from those to the areas that are difficult to resolve at early FWI iterations. In numerical tests, our method demonstrates the ability to obtain convergent FWI results at higher frequencies.