The continuous expansion of LEO satellite constellation has made the relatively scarce gateway stations even more strained. To improve the utilization rate of gateway station antennas, a hybrid clonal selection algorithm is proposed. First, by transforming visible arcs into a task set, the feeder link handover problem is transformed into a task assignment problem, and the corresponding integer programming model is established. Then the antibody is encoded as a set of task assignment vectors, and a decoding method based on the shortest path in directed graphs is proposed in combination with heuristic conflict resolution rules. A threshold parameter is introduced to reduce the computation cost of the decoding method. Furthermore, an adaptive neighborhood selection-based local search algorithm is proposed to enhance local optimization capabilities. Finally, the simulation scenario of feeder link handover in LEO satellite constellation is constructed, and instances of different scales are generated. Experimental results show that the proposed algorithm can quickly converge to the optimal solution for small-scale instances, while showing stronger solving ability and more stable performance than the existing heuristics on large-scale instances, thus verifying its effectiveness.