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Deep neural networks often fail to generalize outside of their training distribution, in particular when only a single data domain is available during training. While test-time adaptation has yielded encouraging results in this setting, we argue that, to reach further improvements, these approaches should be combined with training procedure modifications aiming to learn a more diverse set of patterns. Indeed, test-time adaptation methods usually have to rely on a limited representation because of the shortcut learning phenomenon: only a subset of the available predictive patterns is learned with standard training. In this paper, we first show that the combined use of existing training-time strategies, and test-time batch normalization, a simple adaptation method, does not always improve upon the test-time adaptation alone on the PACS benchmark. Furthermore, experiments on Office-Home show that very few training-time methods improve upon standard training, with or without test-time batch normalization. We therefore propose a novel approach using a pair of classifiers and a shortcut patterns avoidance loss that mitigates the shortcut learning behavior by reducing the generalization ability of the secondary classifier, using the additional shortcut patterns avoidance loss that encourages the learning of samples specific patterns. The primary classifier is trained normally, resulting in the learning of both the natural and the more complex, less generalizable, features. Our experiments show that our method improves upon the state-of-the-art results on both benchmarks and benefits the most to test-time batch normalization.