We consider the $(K,L,M,N)$ multi-access coded caching system introduced by Hachem et al., which consists of a central server with $N$ files and $K$ cache nodes, each of memory size $M$, where each user can access $L$ cache nodes in a cyclic wrap-around fashion. At present, several existing schemes achieve competitive transmission performance, but their subpacketization levels grow exponentially with the number of users. In contrast, schemes with linear or polynomial subpacketization always incur higher transmission loads. We aim to design a multi-access coded caching scheme with linear subpacketization $F$ while maintaining low transmission load. Recently, Cheng et al. proposed a construction framework for coded caching schemes with linear subpacketization (i.e., $F=K$) called non-half-sum disjoint packing (NHSDP). Inspired by this structure, we introduce a novel combinatorial structure named cyclic multi-access non-half-sum disjoint packing (CMA-NHSDP) by extending NHSDP to MACC system. By constructing CMA-NHSDP, we obtain a new class of multi-access coded caching schemes. Theoretical and numerical analyses show that our scheme achieves lower transmission loads than some existing schemes with linear subpacketization. Moreover, the proposed schemes achieves lower transmission load compared to existing schemes with exponential subpacketization in some case.