Transfer texture from one UV map to another?

I textured a wheel, but then thought the rim should probably be a separate object. What I’m now wondering is if I can remove/separate the rim and make it its own object but keep the texture I did on the tire intact.

Is there a way that I can remove the UV islands of the rim from the UV map and make the UV islands from the tire fill the whole map while leaving the texture intact?

For that I thought I could maybe transfer the UV islands with the texture from the tire to a new map so only those are on the UV map. Is that how you would do this?

Already googled about this but I’m still not sure how I would go about this. Thank you.

Create a new UV layout and use the Render / Bake to bake the objects texture from the exiting UVs to the new UV layout