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NO. Given a Riemannian manifold, it might be possible to improve smoothness by changing atlas. The atlas with harmonic functions as coordinates is the best [proved by Samuil Shefel (1979) and rediscovered by Dennis DeTurck and Jerry Kazdan (1981)]. 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.