A heterogeneous system, where small networks (e.g., small cell or WiFi) boost the system throughput under the umbrella of a large network (e.g., large cell), is a promising architecture for the 5G wireless communication networks, where green and sustainable communication is also a key aspect. Renewable energy based communication via energy harvesting (EH) devices is one of such green technology candidates. In this paper, we study an uplink transmission scenario under a heterogeneous network hierarchy, where each mobile user (MU) is powered by a sustainable energy supply, capable of both deterministic access to the large network via one private channel, and dynamic access to a small network with certain probability via one common channel shared by multiple MUs. Considering a general EH model, i.e., energy arrivals are time-correlated, we study an opportunistic transmission scheme and aim to maximize the average throughput for each MU, which jointly exploits the statistics and current states of the private channel, common channel, battery level, and EH rate. Applying a simple yet efficient "save-then-transmit" scheme, the throughput maximization problem is cast as a "rate-of-return" optimal stopping problem. The optimal stopping rule is proved to has a time-dependent threshold-based structure for the case with general Markovian system dynamics, and degrades to a pure threshold policy for the case with independent and identically distributed system dynamics. As performance benchmarks, the optimal power allocation scheme with conventional power supplies is also examined. Finally, numerical results are presented, and a new concept of "EH diversity" is discussed.