Moving the Search UI autocomplete results container

I'm implementing Search UI (React) on a site, and it's working well for the most part, but due to the way the site is structured and styled, I'm running into trouble with existing site elements overlapping and obscuring the autocomplete results box. The layout is fairly complex, so this doesn't appear to be a matter of simply adjusting some z-index attributes.

I've experimented with using an input view on my SearchBox component to alter the structure inside it, but I've reached the conclusion that the autocomplete container likely needs to be fully decoupled from the search form container (for example, appended to the root body element rather than nested 10+ levels deep) for me to be able to position it correctly.

Has anyone come up with an elegant way to do this?