Can we generalize the regular value theorem even beyond the Ehresmann's theorem?

Since there are requests for clarification in the comments, I will present my idea of the proof in the simple case, when the normal bundle to $V$ is trivial. The reasoning fails at one point, maybe someone could help.

By the tubular neighbourhood thm an appropriate neighbourhood of $V$ is diffeomorphic to $B\times V$, where $B$ is a ball in the normal bundle. Since $f$ is submersive, locally we can consider it as a projection and we can (locally!) arrange an $f$-related vector fields corresponding to the standard coordinate vector fields in $B$. By compactness we can assume that $B$ is small enough for these constructions. Then, via partition of unity, we can arrange global (in a neighbourhood of the whole $f^{-1}(V)$) vector fields $f$-related to the standard vector fields in $B$.

If these vector fields establish a diffeomorphism between some neighbourhood $f^{-1}(U)$ of $f^{-1}(V)$ and $B\times f^{-1}(V)$, then - in the product coordinates of respective neighbourhoods - $f$ is given by the formula $f(b,m)=(b,f(m))$.

The thing is that I don't know how to guarantee that these v.f. are "integrable" and how to treat the general case.

Edit: A possible bypass (for integrability) is to fix order of these vector fields and integrate them in that fixed order as in this proof (page 29) of Ehresmann's thm by Peter Petersen. By the IFT and compactness it should work. Am I right?