One goal of the fifth-generation (5G) cellular network is to support much higher data capacity (e.g., 1000 times higher than today), where device-to-device (D2D) communication is one of the key enabling technologies. In this paper, we focus on the D2D relaying functionality to improve cellular downlink throughput. Based on the shortages of the latest relevant work, we propose a new scheme that leverages multi-hop relay-assisted outband D2D communications. First of all, by extending two-hop connection to three-hop connection, our scheme can cut down receiving bit error ratio (BER) of cell edge nodes far away from a cellular base station (BS), which improves cellular downlink throughput. Then, it balances network lifetime and throughput by the proposed ratio of income and expenditure (RIE) metric with respect to remaining energy and throughput. Moreover, it reduces the computational overhead of searching relay and also ensures that the optimal relay is selected by the adjustment of searching scope. Compared with the most relevant work, our scheme outperforms it in terms of throughput, delay, and network lifetime.