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Here is a formal construction, but it will not make you happy:

Fix a $C^\infty$-smoothing $\sigma$. Set $$S(f)=\sigma(f)\cdot(1+h_k)$$ Where $h_k\to0$ and $h_k|\Omega\in [C^{k+1}\backslash C^{k}](\Omega)$ for any open set $\Omega\subset M$.