We consider a communication framework between a swarm of satellites, belonging to a massive low-Earth orbit (LEO) constellation, and a receiving satellite (i.e., the sink). The employed multiple access scheme combines non-orthogonal multiple access (NOMA), where intersatellite links (ISLs) share the same time-frequency resource blocks, and orthogonal multiple access (OMA), where ISLs make use of orthogonal resource blocks. We develop (and numerically test) design strategies of the proposed hybrid NOMA-OMA scheme, which exploit differences in Doppler frequencies (called Doppler frequency diversity) to enhance fairness among the satellites, in comparison to a pure-NOMA scheme, while ensuring a remarkably higher sum-rate capacity compared to a pure-OMA technique.