This paper proposes a novel framework for rate-adaptive semantic communication based on multi-stage vector quantization (VQ), termed \textit{MSVQ-SC}. Unlike conventional single-stage VQ approaches, which require exponentially larger codebooks to achieve higher fidelity, the proposed framework decomposes the quantization process into multiple stages and dynamically activates both stages and individual VQ modules. This design enables fine-grained rate adaptation under varying bit constraints while mitigating computational complexity and the codebook collapse problem. To optimize performance, we formulate a module selection problem that minimizes task loss subject to a rate constraint and solve it using an incremental allocation algorithm. Furthermore, we extend the framework by incorporating entropy coding to exploit non-uniform codeword distributions, further reducing communication overhead. Simulation results on the CIFAR-10 dataset demonstrate that the proposed framework outperforms existing digital semantic communication methods, achieving superior semantic fidelity with lower complexity while providing flexible and fine-grained rate control.