The study of genome rearrangements, the evolutionary events that change the order and strandedness of genes within genomes, presents new opportunities for discoveries about deep evolutionary events. The best software so far, GRAPPA, solves breakpoint and inversion phylogenies by scoring each tree topology through iterative improvements of internal node gene orders. We find that the greedy hill-climbing approach means the accuracy is limited because of multiple local optima. To address this problem, we propose integration GRAPPA with MPME, a string encoding of gene adjacency relationships whose optimal internal node assignments can be determined globally in polynomial time, to provide better initializations for GRAPPA. In simulation studies, the new algorithm yields shorter tree lengths and better accuracy in phylogeny reconstruction.