In an earlier paper, we described a new method for phylogenetic tree reconstruction called the Disk Covering Method, orDCM. This is a general method which can be used with any existing phylogenetic method in order to improve its performance. We showed analytically and experimentally that when DCM is used in conjunction with polynomial time distance-based methods, it improves the accuracy of the trees reconstructed. In this paper, we discuss a variant on DCM, that we call DCM2. DCM2 is designed to be used with phylogenetic methods whose objective is the solution of NP-hard optimization problems. We show that DCM2 can be used to accelerate searches for Maximum Parsimony trees. We also motivate the need for solutions to NP-hard optimization problems by showing that on some very large and important datasets, the most popular (and presumably best performing) polynomial time distance methods have poor accuracy.