In priority-based matching problems, there is a trade-off between efficiency and reducing justified-envy. We show that finding an efficient and justified-envy minimal matching is an NP-hard problem when objects have weak priority rankings. Consequently, we focus on resolving the trade-off in a subclass of efficient mechanisms. First, we consider the class of efficient and strategyproof mechanisms and show that no group-strategyproof mechanism minimizes justified-envy in that class. Then, we restrict attention to the class of hierarchical exchanges and sequential dictatorship mechanisms, which are widely studied in the literature and applied in real-life resource allocation problems. We propose justified-envy minimal mechanisms for each of those classes.