My engagement ring comes with two bands--- one for the top, one for the bottom. When we got engaged, I obviously got the engagement ring. Then, when we found out I was pregnant, FI gave one of the bands to DS to give to me, in symbolism of me, FI and him being a family of 3, regardless of genetics. So, both rings are special to me.
So I wear both the band and the engagement ring. For the last couple of days though, I haven't been wearing them. My hands have been getting so dry, and I actually had a small rash on my finger where the band(s) were so tight. It's fine now, but I have the opposite problem as you do--- the single band fits perfectly, but the engagement ring is tighter. I can start out wearing both in the morning, but I usually have to take one or both off by mid-afternoon.
This has only been within the last week and a half. They were on and fine until that point.