In this paper, we develop a hybrid multiple access (MA) protocol for an intelligent reflecting surface (IRS) aided uplink transmission network by incorporating the IRS-aided time-division MA (I-TDMA) protocol and the IRS-aided non-orthogonal MA (I-NOMA) protocol as special cases. Two typical communication scenarios, namely the transmit power limited case and the transmit energy limited case are considered, where the device's rearranged order, time and power allocation, as well as dynamic IRS beamforming patterns over time are jointly optimized to minimize the sum transmission delay. To shed light on the superiority of the proposed IRS-aided hybrid MA (I-HMA) protocol over conventional protocols, the conditions under which I-HMA outperforms I-TDMA and I-NOMA are revealed by characterizing their corresponding optimal solution. Then, a computationally efficient algorithm is proposed to obtain the high-quality solution to the corresponding optimization problems. Simulation results validate our theoretical findings, demonstrate the superiority of the proposed design, and draw some useful insights. Specifically, it is found that the proposed protocol can significantly reduce the sum transmission delay by combining the additional gain of dynamic IRS beamforming with the high spectral efficiency of NOMA, which thus reveals that integrating IRS into the proposed HMA protocol is an effective solution for delay-aware optimization. Furthermore, it reveals that the proposed design reduces the time consumption not only from the system-centric view, but also from the device-centric view.