Relying on multi-hop communication techniques, aeronautical ad hoc networks (AANETs) seamlessly integrate ground base stations (BSs) and satellites into aircraft communications for enhancing the on-demand connectivity of planes in the air. In this integrated AANET context we investigate the shortest-path routing problem with the objective of minimizing the total delay of the in-flight connection from the ground BS subject to certain minimum-rate constraints for all selected links in support of low-latency and high-speed services. Inspired by the best-first search and priority queue concepts, we model the problem formulated by a weighted digraph and find the optimal route based on the shortest-path algorithm. Our simulation results demonstrate that aircraft-aided multi-hop communications are capable of reducing the total delay of satellite communications, when relying on real historical flight data.