Terrestrial communication networks have experienced significant development in recent years by providing emerging services for ground users. However, one critical challenge raised is to provide full coverage (especially in dense high-rise urban environments) for ground users due to scarce network resources and limited coverage. To meet this challenge, we propose a high altitude platform (HAP)-reserved ground-air-space (GAS) transmission scheme, which combines with the ground-to-space (G2S) transmission scheme to strengthen the terrestrial communication and save the transmission power. To integrate the two transmission schemes, we propose a transmission control strategy. Wherein, the ground user decides its transmission scheme, i.e., switches between the GAS link transmission and the G2S link transmission with a probability. We then maximize the overall throughput and derive the optimal probability that a ground user adopts the GAS transmission scheme. Numerical results demonstrate the superiority of the proposed transmission control strategy.