No St Pat's in Austin... but '71 is about as far back as they seem to go.
Tommy James -Draggin' The Line
Santana -Oye Como Va
Nitty Gritty Dirt Band -Mr. Bojangles
Rolling Stones -Brown Sugar
Bill Withers -Ain't No Sunshine
Led Zeppelin -Stairway To Heaven
Melanie -Brand New Key
Tom Jones -She's A Lady
Janis Joplin -Me And Bobby McGee
ETA: actually 1970 is their limit. But I went back a few months to look at sets and they did "Turkeys" on Thanksgiving, some of which are HFHs...:
Loudon Wainwright III -Dead Skunk
Buckner & Garcia -Pac-Man Fever
Captain & Tennille -Muskrat Love
Baha Men -Who Let The Dogs Out
Starland Vocal Band -Afternoon Delight
Steve Martin -King Tut
Wierd Al Yankovich -Eat It
William Shatner -Lucy In The Sky With Diamonds
William Hung -She Bangs