Space–air–ground integrated network (SAGIN) is considered as an enabler for the sixth-generation (6G) networks. By integrating terrestrial and non-terrestrial (satellite, aerial) networks, SAGIN seem to be a quite promising solution to provide reliable connectivity everywhere and all the time. Their availability can be further enhanced if hybrid free space optical (FSO)/radio frequency (RF) links are adopted. In this paper, the performance of a hybrid FSO/RF communication system operating in SAGIN has been analytically evaluated. In the considered system, a high altitude platform station (HAPS) is used to forward the satellite signal to the ground station. Moreover, the FSO channel model assumed takes into account the turbulence, pointing errors and path losses, while for the RF links, a relatively new composite fading model has been considered. In this context, a new link selection scheme has been proposed that is designed to reduced the signaling overhead required for the switching operations between the RF and FSO links. The analytical framework that has been developed is based on the Markov chain theory. Capitalizing on this framework, the performance of the system has been investigated using the criteria of outage probability and the average number of link estimations. The numerical results presented reveal that the new selection scheme offers a good compromise between performance and complexity.