We had the same problem. Then the nurse suggested to feed the baby before changing him. So my wife would feed him from one side, letting the baby soothe, then we would change him (without crying this time), finally feed on the other side.
Make sure you make the baby burp before changing, or the "handling" may cause reflux.
My son was born in winter, so we also thought it was for the cold. Turns out that wasn't the case!