The Hobbit says
Radagast lived near the Southern border of
Mirkwood, and in my opinion this agrees well enough with where
Elrond's scouts look for him: (. . .)
More technically (gnat-pickery or nitpickery), in
The Hobbit Gandalf says that Radagast
lives near the Southern border of Mirkwood.
Gandalf said this in Third Age 2941, the same year that
Sauron abandoned
Dol Guldur, and if we accept G as being correct here (and why not), that means our favorite brown wizard lived near the south of Mirkwood when Sauron himself was at Dol Guldur,
and, although more generally speaking,
well after a shadow had fallen upon Greenwood -- which began way back in
the Third Age 1050 [and in circa 1100, the Wise discover that
an evil power has made a stronghold at Dol Guldur].
Was Radagast already at
Rhosgobel before the shadow shadowed in TA 1050? In TA 2063
"The power of Dol Guldur grows" and three years later Gandalf goes to DG, and Sauron retreats and hides -- yet in TA 2460 Sauron returns with increased strength to DG, and much later (2851) Gandalf urges an attack on it, but
Saruman overrules him. . .
. . . then, in TA 2939, Sauron's servants are searching the
Anduin near the Gladden
Fields! Kind of "near" Rhosgobel I'd think (here going by
author-published accounts). And then we arrive again at TA 2941, Sauron abandoning Dol Guldur.
The Silvan Elves and Dol Guldur: it's said (Unfinished
Tales,
The Sindarin Princes of the Silvan Elves) that when the Shadow fell upon Greenwood,
Thranduil's folk retreated before it, as it spread ever northward, until at last he established his
realm in the north-east (see also note 14 to
the Disaster of the Gladden Fields).
Okay, so again, where was Raddy during all this? It's said that even
Amroth must have dwelt in growing disquiet ever since Dol Guldur was established, and he dwelt on the other side of the Anduin, and he perished
before 2060 (
"the power of Dol Guldur grows").
In the years between
Bilbo's adventure (2941) and Gandalf meeting Radagast (3018), we have Sauron openly declaring himself, and although he began gathering his power in
Mordor, he also sends three
Nazgul to reoccupy Dol Guldur.
Then there's another meeting of
the White Council, and circa 3000 the shadow of Mordor lengthens. Gandalf and
Aragorn search in Mirkwood in 3009, when it's said (Tale of Years):
"the mountains and all lands Eastward are becoming dangerous."
Anyway, this is but gatherin' stuff for possible consideration (and for me to post something), working within the idea that Rad had possibly moved. Despite the wording already noted, for myself I'm not certain he did, and to my mind the late note (Carrock and so on) need not be counted as "in story" evidence, as this could simply be JRRT "moving" Rhosgobel in an external sense, possibly not remembering what he'd already published.
I also ran across this: in a late note on the
Wizards Tolkien notes that Radagast had remained of good will
"(though he had not much courage)".
Ouch!