I've never had an issue with headspace in my glass carboy. I have a 6-6.5 (can't remember) gallon one that I only use for secondary when needed. I still use a bucket for primary. It's sort of backwards, I know, but it has always worked.
The only beer I've ever oxidized was with a conical that I misused and also had a leaky butterfly valve. Then again, I never use a secondary as merely a waiting vessel, I always add something for extra flavors that usually starts the fermentation again slightly.