It might take a bit of work, but you could add a ¼" by cutting a piece of ¼" ply 4" x 4" finish how you prefer, even if you get wall paper to match, glue this to the spot where you hanger pulled out let it dry. Now you have ½" to attach the hanger. You might have to cut ¼" off the rod is all. You can make the plywood as big as you like, it will just add strength to that portion of the wall.
IF you want to keep it simple, you could by an expanding wall anchor and reattach your hanger again.
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