The hop-limited adaptive routing (HLAR) mechanism and its enhancement (EHLAR), both tailored for the packet-switched non-geostationary (NGEO) satellite networks, are proposed and evaluated. The proposed routing mechanisms exploit both the predictable topology and inherent multi-path property of the NGEO satellite networks to adaptively distribute the traffic via all feasible neighboring satellites. Specifically, both mechanisms assume that a satellite can send the packets to their destinations via any feasible neighboring satellites, thus the link via the neighboring satellite to the destination satellite is assigned a probability that is proportional to the effective transmission to the destination satellites of the link. The satellite adjusts the link probability based on the packet sending information observed locally for the HLAR mechanism or exchanged between neighboring satellites for the EHLAR mechanism. Besides, the path of the packets are bounded by the maximum hop number, thus avoiding the unnecessary over-detoured packets in the satellite networks. The simulation results corroborate the improved performance of the proposed mechanisms compared with the existing in the literature.