The Genomic Median Problem is an optimization problem inspired by a biological issue: it aims at finding the genome organization of the common ancestor to multiple living species. It is formulated as the search for a genome that minimizes some distance measure among given genomes. Several attempts have been made at solving the problem. These range from simple heuristic methods to a stochastic local search (SLS) algorithm that is inspired by a well-known local search algorithm for the satisfiability problem in propositionnal logic, called WalkSAT. The objective of this study is to implement improved algorithmic techniques, particularly ones based on tabu search, in the quest for better quality solutions for large instances of the problem. We have engineered a new high-performing SLS algorithm, extensively tested the developed algorithm and found a new best solution for a real-world case.