The performance of grant-free random access (GF-RA) is limited by the number of accessible random access resources (RRs) due to the absence of collision resolution. Compressive sensing (CS)-based RA schemes scale up the RRs at the expense of increased non-orthogonality among transmitted signals. This paper presents the design of multi-sequence spreading random access (MSRA) which employs multiple spreading sequences to spread the different symbols of a user as opposed to the conventional schemes in which a user employs the same spreading sequence for each symbol. We show that MSRA provides code diversity, enabling the multi-user detection (MUD) to be modeled into a well-conditioned multiple measurement vector (MMV) CS problem. The code diversity is quantified by the decrease in the average Babel mutual coherence among the spreading sequences. Moreover, we present a two-stage active user detection (AUD) scheme for both wideband and narrowband implementation. Our theoretical analysis shows that with MSRA activity misdetection falls exponentially while the size of GF-RA frame is increased. Finally, the simulation results show that about 82% increase in utilization of RRs, i.e., more active users, is supported by MSRA than the conventional schemes while achieving the RA failure rate lower bound set by random access collision.