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Signal reconstruction from the measurements of its Fourier transform magnitude remains an important and difficult problem that occurs in different areas in signal processing. Among all the approaches developed to solve this problem, the iterative transform algorithms are currently the most efficient. However, these algorithms suffer from major drawbacks such as stagnation, slow convergence, and high computational cost that limit their practical application. In this brief, we introduce a wavelet adaptation of the general iterative algorithm where the problem is decomposed into different resolution levels and the image is reconstructed following a coarse-to-fine strategy. We show that the proposed approach can significantly improve the performance of the existing algorithms while dramatically reducing their computational complexity.