Make 'Tag picker' a permanent selection
Instead of display the 'Tag picker' as a modal dialog make it a switchable selection in the 'Add tag' dialog.
The 'Add tag' could display a list of pinned or last used tags.
Below it there could be either a list of all tags/tag combinations or the 'Tag picker'. This selection should be reused, when displaying the 'Add tag' dialog the next time.