There is a growing interest in the transition from 4-step random access to 2-step random access in machine-type communication (MTC), since 2-step random access is well-suited to short message delivery in various Internet of Things (IoT) applications. In this paper, we study a 2-step random access approach that uses code division multiple access (CDMA) to form multiple channels for data packet transmissions with a spreading factor less than the number of channels. As a result, the length of data transmission phase in 2-step random access can be shorter at the cost of multiuser interference. To see how the decrease of the length of data transmission phase can improve the spectral efficiency, we derive the throughput as well as the spectral efficiency. From the results of analysis, we can show that the 2-step CDMA-based random access approach can have a higher spectral efficiency than conventional 2-step approach with orthogonal channel allocations, which means that the performance of MTC can be improved by successfully transmitting more data packets per unit time using CDMA. This is also confirmed by simulation results.