Suppose that I have a smooth map of manifolds $f: M^n \rightarrow D^2\backslash 0$ and $f$ is a submersion, i.e. a fiber bundle over $D^2 \backslash 0$. When is it possible to find another smooth map of manifolds $g: N^n \rightarrow D^2$ such that $M^n \subset N^n$ and $g$ extends $f$? Ideally, I would also want that $N^n \backslash M^n$ has codimension 1 in $N^n$.
Note that I am allowed to pick $N^n$, unlike in the following post Extending a smooth map where $N^n$ is prescribed a priori.
If I don't care that $N$ and $g$ are smooth, then this is always possible. Just take the one point compactification $\hat{M}$ of $M$ and extend the map $f$ continuously to compactification.