Reports have been rife that Alexander had completed a move to Sturrock Park in the last week, but no official word had come from the club.
The 28-year-old midfielder was released by SuperSport United last month, following an unsuccessful 18-month spell that was largely due to an assortment of injuries.
However, Hunt has since confirmed that the Bafana Bafana international has indeed signed, explaining the decision to bring in another central midfielder.
"Yeah, Cole Alexander's here. We've signed him," Hunt says.
"A few [central midfielders] are going, and a few have gone already, so we have to look at the long-term as well," he explained.
Wits parted ways with former Bafana skipper Steven Pienaar at the turn of the year, while Xola Mlambo completed a transfer to Orlando Pirates.
Alexander faces competition with the likes of Thabang Monare, Granwald Scott, Edwin Gyimah, Ben Motshwari and Phumlani Ntshangase for a place in the team.
The Clever Boys visit Golden Arrows in their next Absa Premiership encounter on Saturday afternoon, with kick-off set for 15h30.