Grant-free random access is one of the key elements in the emerging mobile system in which a low-latency communication requirement must be met to support a massive number of machine-type communication (mMTC) devices. The random access success rate of sparse code multiple access (SCMA)-based grant-free (GF) access is mainly determined by factors such as collision in random access resource selection, users' activity, and data misdetection. At the same time, it is challenging to decrease the effect of these factors while reducing signaling overhead. One approach to handle the challenge is to employ a large set of non-orthogonal preambles for active user detection and channel estimation while allowing users' data to be differentiated by their channel even if the same SCMA codebook (CB) is selected. In this paper, we introduce a new approach that maps a user's consecutive symbols to code words from different SCMA CBs in around-robin manner, as opposed to conventional SCMA in which a single codebook (CB) is employed per a user's random access transmission. It is also shown that the proposed multi-codebook SCMA-based GF access scheme outperforms the conventional single-codebook SCMA-based one by overloading of more users(up to 200%) while keeping the same preamble length, i.e., the same level of overhead.