How to render/bake texture onto a new UV layout? (With the texture still intact)

I’ve recently downloaded a free mesh head which I am working with, it already came with a UV map, on which I created a texture but I was just wondering, is it possible to unwrap the UV to a new layout with the texture still intact/fitting to the new layout as well?

I tried the render/bake method, and it successfully creates a texture in the shape of the new UV map layout but it shows as plain white and not as the texture I made for the original UV map, I had uploaded this texture into the textures tab etc before hand so i’m not sure why it’s doing that, does anyone have any idea?

Bake texture from UV set to another