We propose an FFT-based iterative algorithm for solving the Lippmann-Schwinger equation in the context of periodic homogenization of infinitely contrasted linear elastic composites. Our work initially reformulates the Moulinec-Suquet, Eyre-Milton and Monchiet-Bonnet schemes using a residual formulation. Subsequently, we introduce an enhanced scheme, termed Adaptive Eyre–Milton (AEM), as a natural extension of the EM scheme where we optimize a relaxation parameter to minimize the residual. We demonstrate the unconditional linear convergence of the AEM scheme, regardless of initialization and the chosen reference material. The paper further extends the AEM scheme to handle composites with both pores and infinitely rigid inclusions. Practical implementation aspects and illustrative applications in two- and three-dimensional settings are discussed, highlighting the efficiency of the proposed AEM scheme. We particularly emphasize the scheme’s robustness for materials with infinitely large contrasts in elastic properties.