Gandalf's First Dol Guldur Visit + Sauron in the East

[Zigûr]

So I managed to find a quote in support of my "Sauron's hideout in Rhûn" idea, from Professor Tolkien's Last Writings about the objectives of the Blue Wizards:

"Their task was to circumvent Sauron: to bring help to the few tribes of Men that had rebelled from Melkor-worship, to stir up rebellion ... and after his first fall to search out his hiding (in which they failed)"

This certainly implies that Sauron had some kind of top-secret refuge beyond the reach or discovery of the Wise. I suppose logically he must have had somewhere to go after fleeing Dol Guldur.

In the tradition of how he evaded the Valar after the Siege of Utumno, I'm imagining some kind of supervillain-esque underground lair deep beneath the earth...