I've heard from friends that the burgers are just as good at Kuma's Too (Kuma's second location on Diversey), but the lines are short to nothing at off times. I have yet to check it out, though I don't know why.
As long as there isn't a 3:30 Cubs game that day, I would think a Friday at 1 might be worth a shot. Added bonus, if the wait is too long, there is a Gino's East about 3/4 of a mile west at Diversey/Lincoln as a backup.
That place is tiny, isn't it? Or have they expanded? Unless the op is dying for Gino's, Lou Malnati's at Lincoln and Sheffield might be a better option for a quick table.
The best option is the original Gino's East after a visit to the Hancock. No extra time wasted driving, parking, waiting...