Today, unmanned aerial vehicles (UAVs) are very popular in military and commercial applications as well as academic research. Flying ad hoc network (FANET) is a new type of ad hoc network that organizes small drones in an ad hoc form. The movement in 3D space, high mobility, frequent topological changes, restricted resources, and low density are the features of these networks. These features cause serious challenges in designing an appropriate routing approach for these networks. In this paper, a greedy perimeter stateless routing method based on a position prediction mechanism called GPSR+ is proposed for FANETs. This scheme employs a position forecast strategy to approximate the future position of UAVs and modifies the propagation period of hello messages based on this strategy to achieve better adaptability to the dynamic network. In addition, GPSR+ chooses a set of candidate UAVs using a new method called the spherical removal technique to decide on the next-hop UAVs in the routing process. Finally, the most stable UAV toward the destination will be chosen from the candidate set to act as the next-hop. The simulation results show that GPSR+ grows the packet delivery ratio in FANET and extends network longevity because it increases path stability and improves energy consumption. However, the delay in GPSR+ is high.