With the current miniaturization and cost reduction trend, many small imaging satellites are deployed in low Earth orbit (LEO). Efficient scheduling of transmission links is crucial for handling the large volume of imaging data in the time-varying inter-satellite broadband communication networks. This ensures optimal utilization of inter-satellite link resources and gives rise to a novel problem known as the Large-volume LEO satellite Imaging Data Networked Transmission Scheduling Problem (LLSIDNTSP). This problem requires the integrity of the imaging data received on the ground station. A Specific Time-Evolving Graph (STEG) model is introduced to formulate this problem, while a highly effective Sequential Two-Phased Heuristic Algorithm (STPHA) is proposed to solve it. STPHA determines an optimized Contact Plan (CP) with a Contact-Fast Construction Algorithm (CFCA) in the first phase. With this CP as input, it produces an optimal transmission schedule with Linear Programming in the second phase. The CFCA incorporates a novel heuristic called Mission Residual Volume Factor(MRVF), which guides the selection of a proper inter-satellite and satellite-to-ground link at each construction step. Extensive simulation results demonstrate the effectiveness of the proposed STPHA. In particular, it can achieve the optimal solution for small-sized scenarios with one-hundredth of the computing time used by the exact solver CPLEX. Also, it is consistently better than other heuristic algorithms adapted from the literature for large-sized scenarios. Additional experimental analysis is conducted to showcase the effectiveness of the innovative components of STPHA.