Satellite constellations have become a promising solution for future Terabits/s connectivity. However, conventional radio frequency (RF) communication links are unable to achieve such high throughput due to the limited spectrum available. Free-space optical (FSO) communication has recently acquired a growing interest in the satellite communication (Satcom) community as a potential solution. Nonetheless, FSO communication is sensitive to the adverse effects of beam scintillation, beam-wander-induced pointing errors, free-space loss, and weather conditions. To mitigate these effects, we present a new space-air-ground (SAG) FSO transmission system with a strategically deployed high-altitude platform relay. It can effectively remedy the effect of atmospheric impairments on FSO transmission, particularly at high zenith angles. Then, we integrate the proposed SAG-FSO link and traditional space-ground FSO/RF links to further improve system performance and reliability. We carry out a thorough performance analysis of the resulting hybrid SAG-FSO/RF communication systems. Tractable analytical expressions for the symbol error and outage probabilities are derived. The numerical results highlight the significant potential of the proposed, highly innovative Satcom systems over existing solutions. Furthermore, Monte-Carlo simulations are performed to validate the accuracy of the analytical results.