In a standard multiple-choice hashing scheme, each item is stored in one of hash table buckets. The availability of choice in where items are stored improves space utilization. These schemes are often very amenable to a hardware implementation, such as in a router. Recently, researchers have discovered powerful variants where items already in the hash table may be moved during the insertion of a new item. Unfortunately, these schemes occasionally require a large number of items to be moved to perform an insertion, making them inappropriate for a hardware implementation. We show that it is possible to significantly increase the space utilization of multiple-choice hashing schemes by allowing at most one item to be moved during an insertion. Furthermore, our schemes can be effectively analyzed, optimized, and compared using numerical methods based on fluid limit arguments, without resorting to much slower simulations.