In this paper, we investigate the performance of an underlay cognitive hybrid satellite-terrestrial network comprising a primary satellite source with its terrestrial receiver and the secondary transmitter (ST) with its pre-paired users on the ground. Herein, the ST employs a cooperative non-orthogonal multiple access (C-NOMA) scheme in which a nearby NOMA user acts as a relay and detects and forwards the information of the far-away NOMA user during the cooperation phase. Further, the widely adopted shadowed-Rician fading and Nakagami-m fading models are considered for the pertinent hybrid channels. For this overall set-up, we obtain a novel closed-form expression for the outage probability of secondary network in the presence of primary interference power constraint imposed by the adjacent primary satellite network. Finally, our analytical findings are corroborated through various numerical and simulation results.