We present an improved algorithm to solve the near-congruence problem for rigid molecules and clusters based on the iterative application of assignment and alignment steps with biased Euclidean costs. The algorithm is formulated as a quasi-local optimization procedure with each optimization step involving a linear assignment (LAP) and a singular value decomposition (SVD). The efficiency of the algorithm is increased by up to 5 orders of magnitude with respect to the original unbiased noniterative method and can be applied to systems with hundreds or thousands of atoms, outperforming all state-of-the-art methods published so far in the literature. The Fortran implementation of the algorithm is available as an open source library (https://github.com/qcuaeh/molalignlib) and is suitable to be used in global optimization methods for the identification of local minima or basins.