The orthonormal frame bundle of a manifold $M$ depends on the choice a Riemannian metric. However, the orthonormal frame bundles $O^{g}M$ and $O^hM$ for two Riemannian metrics $g$ and $h$ are known to be isomorpic, see for example: Does the $O(n)$ bundle of a manifold depend on the metric? Unfortunately, it is not clear to me how the map $\alpha_{g,h}$ is constructed.
Does anyone know a detailed reference for this statement in the literature? The only thing I was able to find is the article "Spineurs, Operateurs de Dirac et Variations de Metriques" by Jean-Pierre Bourguignon and Paul Gauduchon. But this article also does not explain why the map between the two bundles is smooth.