This paper investigates the problem of data routing in flying ad hoc networks (FANETs) composed of multiple flying nodes, i.e., unmanned aerial vehicles (UAVs), supported by communication platforms. The objective is to exploit the mobility of UAVs in order to establish routing paths and transfer a message between two ground nodes at minimum transmission time. Assuming that the UAVs are already deployed to execute a given primary task, the cooperation of the UAVs in the data transfer process, considered as a secondary task, becomes subject to three conditions. First, the energy consumed by each UAV has to respect the allocated budget for the data routing process. Second, the UAVs cannot move out of the boundaries of a tolerated and well-defined region in order to maintain the operation of the primary task. Finally, the UAVs need to reduce their traveled distances in order to reduce the delay of the transfer. A mixed non-linear integer programming problem determining the routing path and the new locations of the UAVs participating in the data transfer process is formulated. Due to its nonconvexity, we proceed with a deterministic exploratory strategy inspired from the Hooke-Jeeves algorithm to meet the problem goals. Selected numerical results investigate the performance of the proposed solution for different scenarios and compare some of them to those of a metaheuristic approach based on swarm intelligence.