Centralized (re)assignment of workers to jobs is increasingly common in public and private sectors. However, these markets often suffer from distributional problems. We propose a new strategy-proof mechanism that efficiently improves individual and distributional welfare over the status quo. We justify our constructive and practical approach by micro-founding it through the theory of inequality measures in welfare economics. To evaluate the performance of our mechanism, we focus on teacher (re)assignment, where the unequal distribution of experienced teachers across schools is a well-documented concern. Using French data, we demonstrate that our mechanism reduces the teacher experience gap across regions more effectively than benchmarks, including the current mechanism, while providing higher average welfare for teachers.