With the blooming of Internet-of-Things (IoT), we are witnessing an explosion in the number of IoT terminals, triggering an unprecedented demand for ubiquitous wireless access globally. In this context, the emerging low-Earth-orbit satellites (LEO-SATs) have been regarded as a promising enabler to complement terrestrial wireless networks in providing ubiquitous connectivity and bridging the ever-growing digital divide in the expected next-generation wireless communications. Nevertheless, the harsh conditions posed by LEO-SATs have imposed significant challenges to the current multiple access (MA) schemes and led to an emerging paradigm shift in system design. In this article, we first provide a comprehensive overview of the state-of-the-art MA schemes and investigate their limitations in the context of LEO-SATs. To this end, we propose a novel next generation MA (NGMA), which amalgamates the grant-free non-orthogonal multiple access (GF-NOMA) mechanism and the orthogonal time frequency space (OTFS) waveform, for simplifying the connection procedure with reduced access latency and enhanced Doppler-robustness. Critical open challenging issues and future directions are finally presented for further technical development.