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 $\Omega\subset M$.