An unmanned aerial vehicle (UAV) can be used as a relay node to provide a communication service to an isolated user equipment (iUE), which does not have a direct communication link from/to a near base station (BS). UAV-aided relay network (UARN) has been considered to provide a reliable communication service to iUE, and a non-orthogonal transmission scheme is utilized at BS to avoid performance degradation of user equipments inside a cell (cUEs). UE scheduling schemes, which are used for conventional relay networks, cannot maximize the long-term average throughput of cUEs in UARN, where UAV moves from BS to iUE or stays close to iUE. In this paper, a novel UE scheduling scheme is proposed to achieve higher long-term average throughput and fairness simultaneously in UARN, while providing a seamless relay transmission to iUE. Simulation results demonstrate that the proposed UE scheduling scheme outperforms the conventional UE scheduling schemes in UARNs.