In this work, we demonstrate an improved method for iterative phase retrieval with application to coherent diffractive imaging. By introducing additional operations inside the support term of existing iterated projection algorithms, we demonstrate improved convergence speed, higher success rate and, in some cases, improved reconstruction quality. New algorithms take a particularly simple form with the introduction of a generalized projection-based reflector. Numerical simulations verify that these new algorithms surpass the current standards without adding complexity to the reconstruction process. Thus the introduction of this new class of algorithms offers a new array of methods for efficiently deconvolving intricate data.