In recent years, disasters happened in many places, and resulted in power shortage and communication interruption. The Wireless Mesh Networks (WMNs) constituted by Renewable Energy-enabled Base Station (REBS) is regarded as a powerful solution in post-disaster recovery, for its energy harvesting ability and the ready-made facilities. However, this solution needs to address several challenges such as unstable power supply, limited bandwidth and long-term optimization. In this paper, we focus on the issue of energy efficiency when realizing the maximal network throughput in a period of time, by the combination of energy usage and network data distribution. To this end, we firstly analyze the unique features of REBS and its associated network in disaster area. Then a throughput-maximization problem is proposed in order to figure out the maximal network throughput. Based on the maximal value, we count out the most energy-efficiency result while guaranteeing the maximal network throughput. We formulate the proposed model into a two-stage Mixed-Integer Linear Programming (MILP) problem and solve it by branch-and-bound algorithm. Simulation results demonstrate our considered two-stage energy efficient scheme strikes a balance between network throughput and its associated energy consumption, and outperforms the existing schemes.