In this paper, we investigate the outage performance of an uplink (UL) non-orthogonal multiple access (NOMA)-based hybrid satellite-terrestrial relay network (HSTRN), in which two users communicate with the satellite through a decode-and-forward (DF) relay due to the lack of direct link. To provide a comprehensive yet hitherto unexplored outage analysis framework, we consider a more generalized channel model, i.e., the terrestrial and satellite links, respectively, undergo <inline-formula> <tex-math notation="LaTeX">$\alpha -\mu $ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$\kappa -\mu $ </tex-math></inline-formula> shadowed fadings. Under fixed power allocation (FPA) for both multiple access phase and relaying phase, we firstly study three successive interference cancellation (SIC) decoding schemes, of which the first two are refined from existing schemes, while the third one, named as <i>extended SIC (ESIC)</i>, is proposed in this paper to satisfy the quality of service (QoS) decoding criterion, which is shown to offer better performances for both users as compared to the former two SIC schemes. We also propose a novel dynamic power allocation (DPA) scheme for the multiple access phase, termed as <i>enhanced DPA (EDPA)</i>, to overcome both users’ error floor (EF) issue yet provide better user fairness than the conventional DPA in the literature. We then analyze the exact and asymptotic outage performance for three SIC and the <i>EDPA</i> schemes under the generalized channel setting. It is shown that both the proposed <i>ESIC</i> and <i>EDPA</i> can circumvent the EF issue and moreover, each has its own advantage in terms of the diversity order (DO). Our results also reveal that there exists a trade-off between <i>ESIC</i> and <i>EDPA</i>, since the former requires a premise on users’ targets rates to overcome the EF and once this premise is satisfied, no DO degradation will occur, while the latter does not entail such a premise but may face potential DO degradation. Finally, we present numerical results to verify the theoretical analysis, manifest the impacts of key parameters on the system performance, and demonstrate the advantages of our proposed network over other benchmarks.