It is fairly known that given a cover $\{U_i\}_{i \in \mathbb{N}}$ of some smooth manifold $M$ together with smooth transition functions $g_{\alpha \beta} \colon U_{\alpha} \cap U_{\beta} \rightarrow GL_n(\mathbb{R})$ satisfying the cocycle condition $g_{\alpha \beta} g_{\beta \gamma} = g_{\alpha \gamma}$ this defines a smooth vector bundle of rank $n$ on $M$ (upto isomorphism).
Say given another set of such transition functions $h_{\alpha \beta}$ together with homotopies (which are smooth transition functions for every $t$) from the $g_{\alpha \beta}$ to the $h_{\alpha \beta}$ is it true that the smooth vector bundles defined by the $g$ and $h$ are diffeomorphic?
I am aware of such a fact for topological vector bundles, for example Thm. 4.3 and corollaries in Hunsemöller, Fibre Bundles. But I am not convinced that the construction he gives there would be smooth. At least he uses some maximum function (however, I find the proof quite difficult to read and would be happy to see another version of it by another author, even if it should not work for differential bundles).