We present a method for inverting seismic reflections using full-waveform inversion (FWI) with inaccurate starting models. For a layered medium, near-offset reflections (with zero angle of incidence) are unlikely to be cycle-skipped regardless of the low-wavenumber velocity error in the initial models. Therefore, we use them as a starting point for FWI, and the subsurface velocity model is then updated during the FWI iterations using reflection wavepaths from varying offsets that are not cycle-skipped. To enhance low-wavenumber updates and accelerate the convergence, we take several passes through the non-linear Gauss-Seidel iterations, where we invert traces from a narrow range of near offsets and finally end at the far offsets. Every pass is followed by applying smoothing to the cumulative slowness update. The smoothing is strong at the early stages and relaxed at later iterations to allow for a gradual reconstruction of the subsurface model in a multiscale manner. Applications to synthetic and field data, starting from inaccurate models, show significant low-wavenumber updates and flattening of common-image gathers after many iterations.