The next generation of satellite communication systems aims to achieve terabit-per-second throughput. Because of the limited spectrum available, traditional radio frequency (RF) communication links cannot provide such high throughput. Free-space optical (FSO) transmission is a possible alternative that has recently gained increased attention in the satellite community. However, FSO communications are vulnerable to the severe effects of atmospheric turbulence, such as beam-wandering-induced pointing errors and beam scintillation. To successfully remedy such effects, we propose a space-air-ground (SAG) FSO network with a strategically deployed high-altitude platform acting as a relay. We show that such a design can substantially mitigate the effects of atmospheric turbulence, especially when the satellite zenith angle is relatively high. Then we present a novel SAG satellite communication network that integrates the suggested SAG-FSO transmission and conventional hybrid single-hop FSO/RF transmission to improve overall system reliability and performance even further. The numerical results clearly show the potential of the proposed, highly innovative SAG-FSO network architecture.