Full-waveform inversion (FWI) is an iterative nonlinear waveform matching procedure subject to wave-equation constraint. FWI is highly nonlinear when the wave-equation constraint is enforced at each iteration. To mitigate nonlinearity, wavefield-reconstruction inversion (WRI) expands the search space by relaxing the wave-equation constraint with a penalty method. The pitfall of this approach resides in the tuning of the penalty parameter because increasing values should be used to foster data fitting during early iterations while progressively enforcing the wave-equation constraint during late iterations. However, large values of the penalty parameter lead to ill-conditioned problems. Here, this tuning issue is solved by replacing the penalty method by an augmented Lagrangian method equipped with operator splitting (iteratively refined WRI [IR-WRI]). It is shown that IR-WRI is similar to a penalty method in which data and sources are updated at each iteration by the running sum of the data and source residuals of previous iterations. Moreover, the alternating direction strategy exploits the bilinearity of the wave-equation constraint to linearize the subsurface model estimation around the reconstructed wavefield. Accordingly, the original nonlinear FWI is decomposed into a sequence of two linear subproblems, the optimization variable of one subproblem being passed as a passive variable for the next subproblem. The convergence of WRI and IR-WRI is first compared with a simple transmission experiment, which lies in the linear regime of FWI. Under the same conditions, IR-WRI converges to a more accurate minimizer with a smaller number of iterations than WRI. More realistic case studies performed with the Marmousi II and the BP salt models indicate the resilience of IR-WRI to cycle skipping and noise, as well as its ability to reconstruct with high-fidelity, large-contrast salt bodies and subsalt structures starting the inversion from crude initial models and a 3 Hz starting frequency.