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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 $C^\infty$

It not true even in dimension 2. In this case, the best atlas use isotropic coordinates. So the metric is described by a function; if this function is not smooth then no hope.