The smooth structure of an embedded submanifold is unique

By [L1], slice-chart restrictions give a smooth structure on $S$, and [L3] shows that its inclusion is a smooth embedding. Suppose $\mathcal{T}$ is another smooth $k$-manifold structure on the same set for which the inclusion $i:(S,\mathcal{T})\hookrightarrow M$ is a smooth embedding. By [F1], $i$ is a homeomorphism onto $S$ with the same subspace topology and an immersion.

Fix $p\in S$ and a slice chart $\phi :U\to {\mathbb{R}}^m$ around $p$. Choose a chart $\theta :W\to {\mathbb{R}}^k$ on $(S,\mathcal{T})$ at $p$. Since $i$ is an immersion and its image lies in the slice ${\mathbb{R}}^k\times \{0\}$ from [F2], the differential of ${\pi }_k\circ \phi \circ i\circ {\theta }^{-1}$ is an injective endomorphism of ${\mathbb{R}}^k$, hence an isomorphism. By [L2] this map is a local diffeomorphism. Thus the slice coordinates are smooth for $\mathcal{T}$, and conversely $\theta$ is smooth for the slice-chart structure.

Every chart of $\mathcal{T}$ is smoothly compatible with the restricted slice charts, so the identity map between $\mathcal{T}$ and the slice-chart structure is a diffeomorphism. Hence the two smooth structures coincide.