We aim to solve the problem of generating coarse-to-fine skills learning from demonstrations (LfD). To scale precision, traditional LfD approaches often rely on extensive fine-grained demonstrations with external interpolations or dynamics models with limited generalization capabilities. For memory-efficient learning and convenient granularity change, we propose a novel diffusion-state space model (SSM) based policy (DiSPo) that learns from diverse coarse skills and produces varying control scales of actions by leveraging an SSM, Mamba. Our evaluations show the adoption of Mamba and the proposed step-scaling method enable DiSPo to outperform in three coarse-to-fine benchmark tests with maximum 81% higher success rate than baselines. In addition, DiSPo improves inference efficiency by generating coarse motions in less critical regions. We finally demonstrate the scalability of actions with simulation and real-world manipulation tasks. Code and Videos are available at https://robo-dispo.github.io.