In this paper, we study the application of opportunistic non-orthogonal multiple access (NOMA) mode to short-message transmissions with user's power control under a finite power budget. It is shown that opportunistic NOMA mode, which can transmit multiple packets per slot, can dramatically lower the session error probability when W packets are to be transmitted within a session consisting of Ws slots, where Ws >= W and the slot length is equivalent to the packet length, compared to orthogonal multiple access (OMA) where at most one packet can be transmitted in each slot. From this, opportunistic NOMA mode can be seen as an attractive approach for uplink transmissions. We derive an upper-bound on the session error probability as a closed-form expression and also obtain a closed-form for the NOMA factor that shows the minimum possible ratio of the session error probability of opportunistic NOMA to that of OMA. Simulation results also confirm that opportunistic NOMA mode has a much lower session error probability than OMA.