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 $\Omega\subset M$.

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

Fix a $C^\infty$-smoothing $\sigma$. If $f\in [C^{k}\backslash C^{k-1}](M)$ et $$S(f)=\sigma(f)\cdot(1+h_k)$$ Where $(h_k)$ is a fixed sequence of functions such that $h_k\to0$ and $h_k|\Omega\in [C^{k+1}\backslash C^{k}](\Omega)$ for any open set $\Omega\subset M$.