Coded caching is recognized as an effective method for alleviating network congestion during peak periods by leveraging local caching and coded multicasting gains. The key challenge in designing coded caching schemes lies in simultaneously achieving low subpacketization and low transmission load. Most existing schemes require exponential or polynomial subpacketization levels, while some linear subpacketization schemes often result in excessive transmission load. Recently, Cheng et al. proposed a construction framework for linear coded caching schemes called Non-Half-Sum Disjoint Packing (NHSDP), where the subpacketization equals the number of users $K$. This paper introduces a novel combinatorial structure, termed the Non-Half-Sum Latin Rectangle (NHSLR), which extends the framework of linear coded caching schemes from $F=K$ (i.e., the construction via NHSDP) to a broader scenario with $F=\mathcal{O}(K)$. By constructing NHSLR, we have obtained a new class of coded caching schemes that achieves linearly scalable subpacketization, while further reducing the transmission load compared with the NHSDP scheme. Theoretical and numerical analyses demonstrate that the proposed schemes not only achieves lower transmission load than existing linear subpacketization schemes but also approaches the performance of certain exponential subpacketization schemes.