Future wireless networks pose several challenges such as high spectral efficiency, wide coverage, massive connectivity, low receiver complexity, etc. To this end, this letter investigates an overlay-based cognitive hybrid satellite-terrestrial network (CHSTN) combining non-orthogonal multiple access (NOMA) and conventional Alamouti space-time block coding (STBC) techniques. Herein, a decode-and-forward based secondary terrestrial network cooperates with a primary satellite network for dynamic spectrum access. Further, for reliable content delivery and low latency requirements, wireless caching is employed, whereby the secondary network can store the most popular contents of the primary network. Considering the relevant heterogeneous fading channel models and the NOMA-based imperfect successive interference cancellation, we examine the performance of CHSTN for the cache-free (CF) STBC-NOMA and the cache-aided (CA) STBC-NOMA schemes. We assess the outage probability expressions for primary and secondary networks and further, highlight the corresponding achievable diversity orders. Indicatively, the proposed CF/CA STBC-NOMA schemes for CHSTN perform significantly better than the benchmark standalone NOMA and OMA schemes.