In the event of a natural disaster, communications infrastructure plays an important role in organizing effective rescue services. However, the infrastructure-based communications are often affected during severe disaster events such as earthquakes, landslides, floods, and storm surges. Addressing this issue, the article proposes a novel drone-based cellular infrastructure to revive necessary communications for out-of-coverage user equipment (UE) which is in the disaster area. In particular, a matching game algorithm is proposed using one-to-many approach wherein several drone small cells (DSCs) are deployed to match different UEs to reach a stable connection with optimal throughput. In addition, a medium access control framework is then developed to optimize emergency and high priority communications initiated from the rescue workers and vulnerable individuals. The simulation results show that the throughput for the out-of-coverage UEs are significantly improved when the DSCs are deployed in public safety network while the channel access delay is also notably reduced for emergency communications within the affected areas.