There may be an easier way, but here's what I would do. Put the paper you want to use under the rainbow, then use your magic wand tool to select the section of rainbow you want to "fill". Once you have the marching ants around it make sure you have the paper (not the rainbow) highlighted in your layers palette and then go to Layer>New>Layer via cut... then you can delete the portion of paper you don't want to keep. Do this with each section of the rainbow til you have your new rainbow underneath. Then you can delete the original rainbow and merge the new rainbow layers together.
Hope this helps!