Heads up-
The easiest way to do this is via CO.com, login to your account, then do an 'advanced' flight search. Before you click "search," tick the check-box that says "OnePass Reward Upgrade," located next to "nonstop flights only" and near "Continental flights only."

That way, you can do the search, but avoid paying the telephone booking fee, which I was forced to do. BUT BEWARE: make sure you look VERY CLOSELY at your segments, because oftentimes, the results that come up won't be in C class at all, and will have a red note at the bottom, "no availability for flight CO00123."
I find it very aggravating, the way it's set up to charge you the 20,000 miles and $500 for a Transatlantic flight, yet only upgrade your domestic portions in this way. It's sneaky for people who aren't on co.com a lot, anyway.
Good luck finding available seats!!