Mobile communication networks could make a significant difference in rescuing affected people in post-disaster scenarios. However, the existing communication infrastructures tend to be out of service in such scenarios. To solve this issue, Unmanned Aerial Vehicles (UAVs) could be launched as flying base stations to provide the required coverage to Rescue Members (RMs) and allow them to communicate and transmit crucial information through the established links. Meanwhile, with the unpredictable movements of RMs, three serious issues are affecting the deployment of UAVs: (i) the control of their mobility, (ii) their limited energy capacity, and (iii) their restricted communication ranges. Aiming to address these issues, we propose deploying an intelligent connected group of energy-efficient UAVs assisting RMs and providing them communication coverage in the long run. These requirements are satisfied using a deep reinforcement learning strategy to learn the environment dynamics and make good trajectory decisions. Simulation experiments have demonstrated the potential of our framework compared to baseline methods to provide temporary communication networks for emergency response teams during disaster relief missions.