The Jets also released journeyman Billy Cundiff, who went 3-for-3 on field goals in the preseason.
The Jets had a previous interest in Carpenter, and it intensified after Folk missed a 39-yard attempt in overtime last Saturday night against the New York Giants. Cundiff ended up making the game-winner from 32 yards in a strange preseason extra period in which Tom Coughlin called a timeout to ice Folk.
Carpenter, 27, was released this week by the Arizona Cardinals. He lasted only one game as a challenger to Jay Feely; his only attempt, from 26 yards, was blocked. He was released earlier in camp by the Miami Dolphins.
Carpenter isn't worried about having little time to prove himself before the second wave of cuts set for Friday and Saturday.
"It doesn't really matter. It's not my decision," he said. "All I'm going to do is show up, they wanted me up here, I'm here, I'm going to kick the best I can and leave it up to them to make a decision."
Carpenter was a Pro Bowl kicker in 2009. He made 22 of 27 attempts last season for the Dolphins, but he's only 10-for-20 in his career on attempts of 50 yards or more.
Folk has been the Jets' kicker since 2010 and had been having a terrific camp until Saturday night. He made a 54-yarder the previous week against the Jacksonville Jaguars.
Folk doesn't feel slighted that Jets brought in Carpenter.
"I think it's them doing their due diligence," Folk said. "Dan is a good kicker. They're doing their due diligence to make sure they have the best people for this team. That's what I think it's coming down to. I feel like I'm hitting the ball pretty well right now. So I'm pretty confident in what I'm doing. All I can control is what I can do. I'm not worried about what's going on upstairs or what anyone else is doing."
Information from ESPNNewYork.com's Matt Ehalt was used in this report.