In this paper, a two-stage handover (TSH) scheme is proposed for low Earth orbit (LEO) mega-constellation networks (MCNs). Specifically, we model the average elevation angle and service capability to assess the overall satellite link quality in MCNs. In the first stage of TSH, we predict the handover time to enhance the signal-to-noise ratio (SNR) performance. Then, the handover decision is formulated as a multi-objective optimization problem and solved by a combination weighting method. Simulation results illustrate the effectiveness and superiority of the proposed strategy to improve users' quality of experience (QoE).