This paper proposes a tractable solution for integrating non-orthogonal multiple access (NOMA) into massive machine-type communications (mMTC) to increase the uplink connectivity. Multiple transmit power levels are provided at the user end to enable open-loop power control, which is absent from the traditional uplink NOMA with the fixed transmit power. The basics of this solution are firstly presented to analytically show the inherent performance gain in terms of the average arrival rate (AAR). Then, a practical framework based on a novel power map is proposed to associate a set of well-designed transmit power levels with each geographical region for handling the no instantaneous channel state information problem. Based on this framework, the semi-grant-free (semi-GF) transmission with two practical protocols is introduced to enhance the connectivity, which has higher AAR than both the conventional grand-based and GF transmissions. When the number of active GF devices in mMTC far exceeds the available resource blocks, the corresponding AAR tends to zero. To solve this problem, user barring techniques are employed into the semi-GF transmission to stable the traffic flow and thus increase the AAR. Lastly, promising research directions are discussed for improving the proposed networks.