When $M=\mathbb{R}^n$ and $S$ is at least $C^2$, your map $\tilde{s}$ is $C^1$ except at points $p$ where $d_S(p)={1\over\kappa(\tilde{s}(p))}$, where $\kappa$ is the smallest principal curvature of $S$ at $\tilde{s}(p)$. The proof boils down to the inverse function theorem, and I don't think is much harder in the general Riemannian case.

Most references are concerned with the smoothness of the distance function $d_S$, but the smoothness of $\tilde{s}$ comes as a byproduct and is usually buried in the proof. See for example http://www.ams.org/tran/2007-359-12/S0002-9947-07-04260-2/

When $M=\mathbb{R}^n$ and $S$ is at least $C^2$, your map $\tilde{s}$ is $C^1$ except at points $p$ where $d_S(p)={1\over\kappa(\tilde{s}(p))}$, where $\kappa$ is the largest principal curvature of $S$ at $\tilde{s}(p)$. The proof boils down to the inverse function theorem, and I don't think is much harder in the general Riemannian case (After giving it some thought, it seems that the proof in the Euclidean case does not apply readily to the Riemannian case).

Most references are concerned with the smoothness of the distance function $d_S$, but the smoothness of $\tilde{s}$ comes as a byproduct and is usually buried in the proof. See for example http://www.ams.org/tran/2007-359-12/S0002-9947-07-04260-2/