We report on the use of a morphing procedure in a simulated annealing (SA) heuristic developed for set-covering problems (SCPs). Morphing enables the replacement of columns in solution with similar but more effective columns (morphs). We developed this procedure to solve minimum cardinality set-covering problems (MCSCPs) containing columns which exhibit high degrees of coverage correlation, and weighted set-covering problems (WSCPs) that exhibit high degrees of both cost correlation and coverage correlation. Such correlation structures are contained in a wide variety of real-world problems including many scheduling, design, and location applications. In a large computational study, we found that the morphing procedure does not degrade the performance of an SA heuristic for SCPs with low degrees of cost and coverage correlation (given a reasonable amount of computation time), and that it improves the performance of an SA heuristic for problems with high degrees of such correlations.