The space–air–ground Integrated Network (SAGIN) is considered to be a significant framework for realizing the vision of “6G intelligent connection of all things”. The birth of 6G SAGIN also brings many problems, such as ultra-dense dense networks, leading to a decrease in the efficiency of traditional flat network management, and traditional satellite networking solidified network functions, etc. Therefore, combining the 6G SAGIN network with the software-defined network (SDN) is an excellent solution. However, the satellite network topology changes dynamically and the ground user unbalanced distribution leads to the unbalanced load of the SDN controller, which further leads to the increased communication delay and throughput drop, etc. For these problems, a hierarchical multi-controller deployment strategy of an SDN-based 6G SAGIN is proposed. Firstly, the delay model of the network, the load model of the SDN controller, and a loss value as a measure of whether the network delay and controller load are optimal are defined. Then, using the distribution relationship between the SDN controller and the switch node as the solution space, and taking the loss value as the optimization goal, a multi-controller deployment strategy based on the simulated annealing algorithm is used to search for the optimal solution space. Lastly, considering the network topology changes dynamically and the SDN controller imbalance, a switch migration strategy oriented toward load balancing is proposed. We aimed to determine the controller deployment plan through the above two points, balance the controller load, and then improve the network performance. The simulation results show that the controller load is increased by about 7.71% compared to OCLDS, and the running time is increased by 17.7% compared to n-k-means.