The unmanned aerial vehicle base station (UAV-BS) plays an important role and has unique advantages in many networking application. The current base station placement and coverage methods are mainly aimed at static networks (assuming that the network nodes are in a static state) and seek global coverage of the UAV-BS to the ground terminals. However, in practical applications, the network nodes are mobile, and global coverage is sometimes difficult to achieve. This paper shows a new solution for inadequate area coverage and random movement of ground terminals. The UAV-BS mobility strategy is used, combined with the delay-tolerant service request method. Using a test platform based on the ONE simulator, the proposed method is simulated and analyzed, and the results confirm that the call service request success rate is significantly improved. In the simulation environment, using the delay-tolerant service request and certain mobility strategies brings an average increase of 5.9% and 10.82% in the call service request success rate, and even increases by more than 20% in some mobile scenarios. Simultaneously, the delay distribution of the call service requests indicates that the proportion of call services with real-time request is optimized, which shows the positive coverage effect of the new method.