This paper presents a novel achievable scheme for coded caching systems with $N$ files and $K$ users, specifically when $N \leq K$. This new scheme employs linear coding both during the placement phase - where cache contents are linear combinations of files from the library - and the delivery phase. The multi-step delivery phase enables users to decode the cached coded content and eliminate interference effectively. In the small memory regime, the proposed scheme outperforms existing methods, particularly when $K$ and $N$ values are similar, it maintains manageable sub-packetization levels, and operates over a finite field of size $3$ regardless of the system parameters.