We propose a new local algorithm for the thermalization of n-vector spin models, which can also be used in the numerical simulation of SU(N) lattice gauge theories. The algorithm combines heat-bath (HB) and micro-canonical updates in a single step - as opposed to the hybrid overrelaxation method, which alternates between the two kinds of update steps - while preserving ergodicity. We test our proposed algorithm in the case of the one-dimensional 4-vector spin model and compare its performance with the standard HB algorithm and with other HB-inspired algorithms.