Great question, and one I was recently asking myself.
My gut instinct is it first appeared in the Craig Cooper call (the one from Jeff's son), also around '02-ish.
Weird trivia from the other direction: I don't believe the Warden of Newbridge Correctional, Bill Rutliffe, ever used "the hut" once during his 40-minute call this year.
I'm also curious as whether Tom or Jon came up with "the hut." Obviously, Jon gets to say all the Wursterisms, but they both script the calls. Then again, Jon is always Tweeting about "Locked Up Abroad," so maybe it's a prison culture thing he's found funny for a while now.