The multi-criteria minimum spanning tree (mc-MST) problem is a typical NP-hard problem. An algorithm to enumerate the set of Pareto optimal spanning trees on some mc-MST instances was put forward by Zhou and Gen, but it does not guarantee returning all the Pareto optimal solutions. To settle this problem, an improved algorithm is developed and also proved to be able to find all the true Pareto optimal solutions in this paper. The new algorithm adds some conditions in the elimination of subtrees. Simulation results show that the new algorithm can find all the Pareto optimal solutions and also show that the new algorithm has potential usage in practice.