Because everyone in his family had worked in show business at some point, Hunt always felt he would end up in entertainment as well. While in middle school and high school, he put on puppet shows for local children, and he was a fan of the then fledgling Muppets from a young age. After high school graduation, and a four-month stint of doing weather reports at a local radio station, Hunt pursued a meeting with Jim Henson.
After being hired to work on Sesame Street, Hunt mostly performed background characters in early specials. One of his first major performances was as Taminella Grinderfall in The Frog Prince, puppeteering the character while Jerry Juhl performed the voice.
He also did Scooter, and shared Miss Piggy with Frank Oz until the final quarter of the first season of The Muppet Show. Hunt performed many characters on Sesame Street, including Forgetful Jones, Placido Flamingo, Don Music, Gladys the Cow, Sully, and an early version of Elmo. On Fraggle Rock, Hunt's main role was the performing the facial expressions and voice of Junior Gorg, but he also performed Gunge (one of the Trash Heap's barkers), and several one shot or minor characters.