For audio source separation applications, it is common to estimate the magnitude of the Time-Frequency (TF) representation of each source. In order to recover a time-domain signal from a spectrogram for instance, it then becomes necessary to recover the phase of the corresponding complex-valued Short-Time Fourier Transform (STFT). Most authors in this field choose a Wiener-like filtering approach which boils down to using the phase of the original mixture. In this paper, a different standpoint is adopted. Many music events are partially composed of slowly varying sinusoids and the STFT phase increment of those frequency components takes a specific form. This allows phase recovery by an unwrapping technique once a short-term frequency estimate has been obtained. Herein, a whole iterative source separation procedure is proposed which builds upon these results. It is tested on a variety of data, both synthetic and realistic, and also with different source separation scenarios, oracle or non oracle. In terms of SIR, SAR and SDR, the method achieves better performance than consistency-based approaches. To complete the experimental analysis, sound examples are provided which allow the reader to assess the interest of the method regarding the improvement of sound quality.