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Abstract Gene trees can be different from the species tree due to biological processes and inference errors. One way to obtain a species tree is to find one that maximizes some measure of similarity to a set of gene trees. The number of shared quartets between a potential species tree and gene trees provides a statistically justifiable score; if maximized properly, it could result in a statistically consistent estimator of the species tree under several statistical models of discordance. However, finding the median quartet score tree, one that maximizes this score, is NP-Hard, motivating several existing heuristic algorithms. These heuristics do not follow the hill-climbing paradigm used extensively in phylogenetics. In this paper, we make theoretical contributions that enable an efficient hill-climbing approach. Specifically, we show that a subtree of sizemcan be placed optimally on a tree of sizenin quasi-linear time with respect tonand (almost) independently ofm. This result enables us to perform subtree prune and regraft (SPR) rearrangements as part of a hill-climbing search. We show that this approach can slightly improve upon the results of widely-used methods such as ASTRAL in terms of the optimization score but not necessarily accuracy.