This paper develops a novel multiple access (MA) protocol for an intelligent reflecting (IRS) aided uplink transmission network by incorporating the IRS-aided time-division MA (I-TDMA) protocol and the IRS-aided non-orthogonal MA protocol (I-NOMA) protocol as special cases. Under the proposed IRS-aided hybrid MA (I-HMA) framework, a sum transmission delay minimization problem is formulated by jointly optimizing the device's rearranged order, time and power allocation, as well as dynamic IRS beamforming patterns over time subject to the transmit power constraints. An analytical comparison for the minimum sum delay of the proposed I-HMA protocol, I-TDMA as well as I-NOMA is provided, and a computationally efficient algorithm is proposed to obtain the high-quality solution to the optimization problem. Simulation results reveal that the proposed protocol can significantly reduce the sum delay by combining the additional gain of dynamic IRS beamforming with the high spectral efficiency of NOMA, which indicates the effectiveness of integrating IRS into the proposed HMA protocol.