In the post-fifth generation (5G) era, escalating user quality of service (QoS) strains terrestrial network capacity, especially in urban areas with dynamic traffic distributions. This paper introduces a novel cooperative unmanned aerial vehicle relay-based deployment (CUD) framework in satellite air-ground integrated networks (SAGIN). The CUD strategy deploys an unmanned aerial vehicle-based relay (UAVr) in an amplify-andforward (AF) mode to enhance user QoS when terrestrial base stations fall short of network capacity. By combining low earth orbit (LEO) satellite and UAVr signals using cooperative diversity, the CUD framework enhances the signal to noise ratio (SNR) at the user. Comparative evaluations against existing frameworks reveal performance improvements, demonstrating the effectiveness of the CUD framework in addressing the evolving demands of next-generation networks.