It *may* be a reaction to the patch adhesive, but it also could be trapped irritants under the patch. The next time you change one, try washing the skin with soap and water and then cleaning it thoroughly again with rubbing alcohol before applying the new patch. If you continue to have issues, you may need to talk to your doctor about alternatives, including other brands/styles of patch. You can also go to twice weekly changes even with the same brand - you just go down to a patch that's half the original dosage twice as often. (I'm not talking about specific dose amounts, but my wife did that because the once a week patches were giving her horrific mood swings.)
I researched this back when my wife was having issues. The manufacturers of generic Climara all use the same adhesive, but it is NOT the same as the brand name's (I assume it's cheaper?) and the other brand name patches also use different adhesives. So you can also try switching brands to see if you react better to a different one. I hear there's a generic Vivelle due out soon, too, which very likely will be yet another distinct adhesive, and also obviously cheaper.