Abstract Solvers for the Maximum satisfiability (MaxSAT) problem find an increasing number of applications today. We focus on improving MaxHS—one of the most successful recent MaxSAT algorithms— via SAT-based preprocessing. We show that employing SAT-based preprocessing via the so-called labelled CNF (LCNF) framework before calling MaxHS can in some cases greatly degrade the performance of the solver. As a remedy, we propose a lifting of MaxHS that works directly on LCNFs, allowing for a tighter integration of SAT-based preprocessing and MaxHS. Our empirical results on standard crafted and industrial weighted partial MaxSAT Evaluation benchmarks show overall improvements over the original MaxHS algorithm both with and without SAT-based preprocessing.