With the return to health for Schneider, Wedgewood was traded by the Devils to the Arizona Coyotes in exchange for a fifth-round pick in the 2018 NHL entry draft on October 28, 2017.

[17] Acquired by the Dallas Stars as they sought depth in goal during a push for the playoffs, and with injuries to their primary goaltender, Jake Oettinger, Wedgewood was given an opportunity to make an immediate impact.

One of his standout moments came on April 12, 2022, when he earned a 1-0 shutout victory against the two-time defending Stanley Cup champion and former club, the Tampa Bay Lightning, making 25 saves in the process.

During this stretch, Wedgewood excelled, going 7-1-1 while delivering several standout performances, including an overtime save against Elias Pettersson of the Vancouver Canucks that immediately led to a game-winning goal by Matt Duchene on December 21, 2023.

After three successful seasons with the Stars, having established himself in the NHL as a backup, Wedgewood left as a free agent and was signed to a two-year, $3 million contract with the Nashville Predators on July 1, 2024.

[22] With an injury to starting goaltender Juuse Saros, Wedgewood made his debut with the Predators in the 2024–25 season opening 4–3 defeat against former club, the Dallas Stars, on October 10, 2024.

[23] Wedgewood collected 1 win through 5 appearances with the Predators through the opening two months of the campaign before he was traded to divisional opponent the Colorado Avalanche, in exchange for Justus Annunen and a sixth-round selection in the 2025 NHL entry draft on November 30, 2024.

[24][25] Wedgewood made his debut in relief for the Avalanche, posting his 50th career win in helping Colorado overcome a 0–4 deficit en route to a 5–4 regulation victory over the Buffalo Sabres on December 3, 2024.

His standout performance came in a 3-2 victory against the United States, where he made 30 saves to secure the win and help Canada finish atop Group B with a perfect 4-0 record.

Entering the net in the semi-finals against Russia, Wedgewood was pulled in a 5-6 defeat, marking the first time in 11 year Team Canada missed the gold medal match.