Suppose we have a smooth (intrinsic) $d$-manifold $M$ (so $M$ is topological space, 2nd countable, Hausdorff, locally $d$-Euclidean, and there are charts $\{\phi_i:\mathbb R^d \to M\}_{i\in I}$ (homeomorphisms to their images) whose transition maps $\phi_j^{-1} \circ \phi_i$ are smooth). By the topological Whitney embedding theorem, we know that there is a closed subset $X\subseteq \mathbb R^N$ s.t. $X$ is a (extrinsic) $d$-manifold homeomorphic to $M$. ($N$ can be taken to be as low as $2d+1$ by dimension theory; in fact $2d$ by substantially more difficult arguments.) That is, $X$ is locally the graph of some (injective) continuous functions, i.e. there are some (continuous) coordinate charts $\{\psi_a:\mathbb R^d \to X\}_{a\in A}$ (homeomorphisms to their images).
The smooth Whitney embedding theorem I think tells us that we can upgrade the $\psi_a$ to be smooth functions.
Question: Is it possible to perform this upgrade by relatively easy arguments?
EDIT: the reason I even dared to believe this was possible was because of this comment by Tom Goodwillie in the above linked MO post
The idea of the $2n+1$ theorem is that "most" smooth maps from a compact smooth $n$-manifold to a $2n+1$-manifold are embeddings -- in particular every map is smoothly homotopic to an embedding, by an arbitrarily short homotopy. This, coupled with the relatively easy result that every continuous map between smooth manifolds is homotopic to a smooth map, implies that every map is homotopic to a smooth embedding. In particular every $n$-manifold embeds in $\mathbb R^{2n+1}$.
It is apparently "relatively easy" to make a continuous map into a smooth map (by a homotopy), so perhaps it is possible to make a continuous embedding $\hookrightarrow \mathbb R^N$ into a smooth embedding $\hookrightarrow \mathbb R^N$, i.e. the desired "upgrading" mentioned above.
As an explicit example, if $M$ is say a $2$-sphere in $\mathbb R^3$ (with smooth atlas/coordinate charts $\{\phi_i\}_{i\in I}$), and $X \subseteq \mathbb R^3$ is a "crumpled" version of the sphere where $\Phi: M \to X$ is a homeomorphism, I think $(X; \{\Phi \circ \phi_i \}_{i\in I})$ forms a smooth manifold (the transition maps are exactly the same as in the original atlas).
The meat of the desired "upgrade" is to somehow recognize (or somehow see encoded in the smooth transition maps) that the crumpled sphere $X$ can be "smoothed" to an actually smooth sphere $M$.