Caching is crucial in hybrid satellite-terrestrial networks to reduce latency, optimize throughput, and improve data availability by storing frequently accessed content closer to users, especially in bandwidth-limited satellite systems, requiring strategic Medium Access Control (MAC) layer. This paper addresses throughput optimization in satellite-terrestrial integrated networks through opportunistic cooperative caching. We propose a joint probing and scheduling strategy to enhance content retrieval efficiency. The strategy leverages the LEO satellite to probe satellite-to-ground links and cache states of multiple cooperative terrestrial stations, enabling dynamic user scheduling for content delivery. Using an optimal stopping theoretic approach with two levels of incomplete information, we make real-time decisions on satellite-terrestrial hybrid links and caching probing. Our threshold-based strategy optimizes probing and scheduling, significantly improving average system throughput by exploiting cooperative caching, satellite-terrestrial link transmission, and time diversity from dynamic user requests. Simulation results validate the effectiveness and practicality of the proposed strategies.