The Lightning signed Miettinen to a two-year contract Monday, but had to wait for him to clear waivers. The Jets used that opportunity to obtain the 31- year-old, who started this season in Russia's Kontinental Hockey League.
Miettinen has played in 472 games in seven NHL seasons with Dallas and Minnesota. He has posted 89 goals and 123 assists.
Hainsey, who had been on the IR with a lower body injury, has appeared in only five games this season and hasn't played since November 23.