Computing locally injective mappings with low distortion in anefficient way is a fundamental task in computer graphics. By re-visiting the well-known MIPS (Most-Isometric ParameterizationS)method, we introduce an advanced MIPS method that inherits thelocal injectivity of MIPS, achieves as low as possible distortionscompared to the state-of-the-art locally injective mapping techniques,and performs one to two orders of magnitude faster in computinga mesh-based mapping. The success of our method relies on twokey components. The first one is an enhanced MIPS energy functionthat penalizes the maximal distortion significantly and distributesthe distortion evenly over the domain for both mesh-based and mesh-less mappings. The second is a use of the inexact block coordinatedescent method in mesh-based mapping in a way that efficiently min-imizes the distortion with the capability not to be trapped early bythe local minimum. We demonstrate the capability and superiority ofour method in various applications including mesh parameterization,mesh-based and meshless deformation, and mesh improvemen