Kovalainen won the 2008 Hungarian Grand Prix for McLaren and earned three more podium finishes in an F1 career that tallied 111 starts between 2007 and 2013.
He subsequently switched to racing in Japan and won the Super GT title at the second attempt in 2016, a feat managed by Button two years later.
Interview: How Super GT helped heal Kovalainen’s F1 "scars"
The 2004 Race of Champions winner called time on his Super GT exploits at the end of 2021 and focused on rallying last season, scooping the Japanese rally championship and finishing tenth overall in a Skoda Fabia R5 WRC2 car on the World Rally Championship's return to the country in November.
The Finn now links up with Button, his replacement at McLaren for 2010, to join the XE field for its third season of competition which begins in Saudi Arabia later this month.
He will pair up with Norwegian Hosas, who managed a best finish of third in the first Island X Prix in Sardinia last season alongside Kevin Hansen.
"I'm really excited to join JBXE in the Extreme E series in Saudi Arabia," said Kovalainen, who is also contesting the Finnish Rally Championship this year.
"It's a great championship with great teams and drivers so I'm looking forward to competing with them.
“It'll be something new again in my career. After my Formula 1 stint I've done GTs as well as some rallying and I'm always looking for new challenges and looking forward to giving this a try.
"It'll be a steep learning curve no doubt but I'm up for it and hopefully we'll get up to speed quickly."
Button said it had been a "no brainer for the team to approach" Kovalainen after his switch to rallying.
"I’m over the moon he’s agreed to join the team and equally delighted to be keeping Hedda after she has continued to make great progress in the car, she fully deserves her place," he said.
Hosas added: "To have the support and expertise of both Jenson and Heikki, two ex F1 drivers, will help me to continue my development as a driver."
Hansen meanwhile has switched to Veloce to pair up with 2021 XE champion Molly Taylor, while new outfit Carl Cox Motorsport has signed two-time DTM champion Timo Scheider and former Veloce driver Christine Giampaoli Zonca.
