We've been talking about splitting the integration of the flyout into different selectors/blocks and place them outside the flyout. In fact, the concept of the flyout may not apply anymore for Sphinx theme after moving this information outside the current flyout -- leaving the concept of flyout only to the one injected by Read the Docs addons.

Using CPython as an example, the version and language selectors could be placed at the top left, below the project's logo/name:

Besides, the "On Read the Docs" section from the original flyout could be moved to the footer:

These are just examples, but I think it reflects how theme authors will integrate with us in a better way without the following the flyout concept.

Related: readthedocs/addons#51