NO. Given a Riemannian manifold, it might be possible to improve smoothness by changing atlas. It is proved by Shefel, that the atlas with harmonic functions as coordinates is the best. But, the obtained metric might be worse than $C^\infty$.
There is no local-global issue here, harmonic atlas is defined locally and it is the best one globally. So you get problems starting with dimension 2.
Anton Petrunin
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