For UA, even with the 1K desk, I always look up every alternative I can think of as I'm going through the process of connecting to a phone agent. Even though those are generally good, experienced agents, they may not have the entire permutation of Star Alliance possibilities to *my* destination memorized.

I got lucky at ORD last year. I had misconnected - very late on a snowy night, prepared to spend the night. Everything I could think of had departed or had canceled already. Just as I was about to give up, I noticed a tail off in the distance - that of an SAS Airbus. I took a guess: I asked the agent any way to get on that Airbus and route through wherever it was headed. Sure enough - it had seats available (last row of coach!) and departed Chicago at midnight for CPH. I made it to my destination 5 hours late instead of 24. ^