This paper investigates uplink transmission rate enhancement in space-air-ground integrated networks (SAGIN) by jointly designing precoders for a multi-antenna ground user and an unmanned aerial vehicle (UAV). Assuming a stationary relative position between the UAV and the satellite, we propose two joint precoding designs to optimize the uplink transmission rate while considering the practical finite signaling. We introduce an alternating iteration optimization approach (AIOA) when accurate channel state information (CSI) of the Rician link from the ground user and UAV is available. Additionally, we account for the statistical CSI induced by multi-path effects in various terrestrial environments, and derive a new closed-form expression for the uplink transmission rate. Building on this, a convex optimization framework is formulated by vectoring the optimization matrix and introducing auxiliary variables to tackle the non-convexity of the problem. Then, the AIOA is further adopted to jointly optimize the ground user and UAV precoders, significantly reducing computational complexity. Simulation results confirm the efficiency of the proposed AIOAs improving uplink transmission rates in SAGIN.