Degrees of freedom (DoF) of the three-user multiple-input multiple-output (MIMO) broadcast channel (BC) with delayed CSIT was derived for most antenna configurations except for the case of 2N < M < 3N, where transmitter has M antennas and each receiver has N antennas. In this paper, for that problem, we propose an effective scheme for acquiring a higher achievable DoF than the value via existing methods. In the initial transmission phase, we transmit more data symbols than the amount that the receivers can instantaneously decode. Then, we generate auxiliary symbols for decoding the data symbols. Specifically, our scheme introduces an integrated design for the generation of auxiliary symbols. As a result, a higher achievable DoF, i.e., 12MN/(7M + 2N), can be achieved for specific antenna 7M +2N configurations, where 2N < M < 2.5N.