While mutation rates can vary markedly over the residues of a protein, multiple sequence alignment tools typically use the same values for their scoring-function parameters across a protein’s entire length. We present a new approach, called adaptive local realignment, that in contrast automatically adapts to the diversity of mutation rates along protein sequences. This builds upon a recent technique known as parameter advising that finds global parameter settings for aligners, to adaptively find local settings. Our approach in essence identifies local regions with low estimated accuracy, constructs a set of candidate realignments using a carefully-chosen collection of parameter settings, and replaces the region if a realignment has higher estimated accuracy. This new method of local parameter advising, when combined with prior methods for global advising, boosts alignment accuracy as much as 26% over the best default setting on hard-to-align protein benchmarks, and by 6.4% over global advising alone. Adaptive local realignment, implemented within the Opal aligner using the Facet accuracy estimator, is available at facet.cs.arizona.edu.