In the field of model order reduction for frequency response problems, the minimal rational interpolation (MRI) method has been shown to be quite effective. However, in some cases, numerical instabilities may arise when applying MRI to build a surrogate model over a large frequency range, spanning several orders of magnitude. We propose a strategy to overcome these instabilities, replacing an unstable global MRI surrogate with a union of stable local rational models. The partitioning of the frequency range into local frequency sub-ranges is performed automatically and adaptively, and is complemented by a (greedy) adaptive selection of the sampled frequencies over each sub-range. We verify the effectiveness of our proposed method with two numerical examples.