The low-orbit mega-constellation network (LMCN) is an important part of the space-air-ground integrated network system. An effective satellite-ground interconnection design can result in a stable constellation topology for LMCNs. A naïve solution is accessing the satellite with the longest remaining service time (LRST), which is widely used in previous designs. The Coordinated Satellite-Ground Interconnecting (CSGI), the state-of-the-art algorithm, coordinates the establishment of ground-satellite links (GSLs). Compared with existing solutions, it reduces latency by 19% and jitter by 70% on average. However, CSGI only supports the scenario where terminals access only one satellite, and cannot fully utilize the multi-access capabilities of terminals. Additionally, CSGI's high computational complexity poses deployment challenges. To overcome these problems, we propose the Classification-based Longest Remaining Service Time (C-LRST) algorithm. C-LRST supports the actual scenario with multi-access capabilities. It adds optional paths during routing with low computational complexity, improving end-to-end communications quality. We conduct our 1000 s simulation from Brazil to Lithuania on the open-source platform Hypatia. Experiment results show that compared with CSGI, C-LRST reduces the latency and increases the throughput by approximately 60% and 40%, respectively. In addition, C-LRST's GSL switchings number is 14, whereas CSGI is 23. C-LRST has better link stability than CSGI.