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Atomistic simulations of the thermodynamic properties of magnetic materials rely on an accurate modeling of magnetic interactions and an efficient sampling of the high-dimensional spin space. Recent years have seen significant progress with a clear trend from model systems to material-specific simulations that are usually based on electronic-structure methods. Here we develop a Hamiltonian Monte Carlo framework that makes use of auxiliary spin dynamics and an auxiliary effective model, the temperature-dependent spin-cluster expansion, in order to efficiently sample the spin space. Our method does not require a specific form of the model and is suitable for simulations based on electronic-structure methods. We demonstrate fast warm-up and a reasonably small dynamical critical exponent of our sampler for the classical Heisenberg model. We further present an application of our method to the magnetic phase transition in bcc iron using magnetic bond-order potentials.