Efficient uplink access is a keystone for the successful deployment of machine-type communication (MTC) that enables the promising Internet of Things (IoT). In this article, we introduce a communication framework that provides uplink access for MTC while avoiding both collisions and large signaling. Particularly, we propose a two-stage uplink access technique that combines both fast-uplink grant and nonorthogonal multiple access (NOMA). Using the fast grant, the base station (BS) schedules the devices without sending scheduling requests. Afterward, NOMA facilitates the grant sharing where pairing is done in a distributed manner to reduce signaling overhead. In addition, NOMA plays a major role in decoupling the challenges associated with fast grant, namely, active set prediction and optimal scheduling. In massive networks, full information gathering at the BS is impractical. Hence, multiarmed bandit (MAB) learning is adopted to schedule the fast grant devices. We devise an abstraction model for the source traffic predictor needed for a fast grant such that the prediction error can be evaluated. Consequently, the performance of the proposed scheme is analyzed in terms of average resource wastage and outage probability. The simulation results show the effectiveness of the proposed method in saving the scarce resources while verifying the analysis accuracy. In addition, the results show that the proposed scheme can easily attain the impractical optimal OMA performance, in terms of the achievable rewards, at an affordable complexity. Furthermore, the ability of the proposed scheme in picking quality MTC devices (MTDs) with strict latency is also proved.