I can't answer that for sure, but if we had "space available" on a flight, we would carry freight on a contract basis (usually on back-hauls where you're basically flying MT, simply to reposition the plane). Likewise, we would contract with other carriers to carry our freight if we were over capacity or facing delays.
There are a lot of "behind the scenes" deals that go on. For example:
TOLOH normally picks up its air from DTW (Detroit Metro). Sometimes during peak, LOUKY would by plan (don't know if they still do) fly an overflow bird directly into TOL (Toledo Express) with extra cans on it to get them out earlier. Since UPS doesn't have any facilities at TOL, the bird would go to the FDX complex, they would unload our cans and we would send a 4-hut trailer out to pick them up there. The bird would then continue on up to DTW with nothing but Metro Detroit pkgs on it.
Does UPS carry USPS mail and packages? If it's profitable, and doesn't compromise service to our customers, it wouldn't suprise me in the least.