In this paper, a hybrid non-orthogonal multiple access (NOMA) framework for the simultaneous transmitting and reflecting reconfigurable intelligent surface (STAR-RIS) enhanced cell-edge communication is investigated. Specifically, one transmitted user and one reflected user are paired as one NOMA-pair, while multiple NOMA-pairs are served via time division multiple access (TDMA). The objective is to maximize the minimum downlink rate by jointly optimizing the user pairing, decoding order, passive beamforming, power and time allocation. A novel two-layer iterative algorithm is proposed to solve the highly coupled problem. Simulation results show that: 1) the proposed framework outperforms the conventional reflecting-only-RIS-based and the OMA-based frameworks; 2) the beamforming design and power allocation dominate the achieved performance; 3) increasing the number of passive elements and shortening the distance between BS and STAR-RIS are two effective ways to further improve the performance.