Welcome to the clean hoop club!

I have used Goo-Gone to remove labels from some items, but always have to wash thoroughly afterward to get rid of the oily residue and the odor. I am not certain I'd like it in my sewing room around fabric.

That stuff is great for all kinds of things. I can't always find it, though. Another good product is WD-40, for the same kind of clean-up. But WD-40 is a lubricant, so it will leave an oily residue that has to be cleaned up.

I like that stuff, too - hard to find however. *4U

Eucalyptus oil removes glued on labels from glass, china and of course my hoops get a good clean with the oil as well.
It just takes away all the hand grease and any other residue that is on the hoops.

Smells great so your sewing room will smell fresh.

I would not recommend it for fabric, because it is an oil base product and might stain.