The Encyclopedia of Arda speculates that after Gandalf passed, Gwaihir recovered Glamdring, and at that same time, I would assume he retrieved Gandalf's staff too. Pure speculation, but since Gandalf had Glamdring when he came back, it must've been retrieved somehow, and it would seem strange for his sword to be retrieved while his staff was left.
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...take counsel with thyself, and remember who and what thou art.
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