The need for spatial adaptivity in simulations of rarefied gases arises because different subdomains within a rarefied gas domain often require varying levels of modeling complexity. Different moment models are effective at describing rarefied flows with respective degrees of complexity in each subdomain. However, currently there is no operational spatially adaptive method to efficiently couple moment models in a single simulation. This paper presents a Spatially Adaptive Moment Model (SAMM) for simulating linear hierarchical moment models. A padded buffer cell approach is proposed to couple these varying-order moment models. Numerical shock tube simulations demonstrate that the proposed model achieves accurate results comparable to a high-order model while providing a computational speedup corresponding to the reduction of variables.