Hutton left after the first period, when he allowed one goal on 11 shots.
Karel Vejmelka
made 26 saves in relief.
Hutton is 0-2-0 with a 7.76 goals-against average and .741 save percentage in three starts this season. The 35-year-old signed a one-year contract with Arizona on July 28 after three seasons with the Buffalo Sabres.
Timmins, a defenseman, left after being checked into the boards by Panthers defenseman Radko Gukas in the second period. Timmins has no points in six games this season after being traded to Arizona by the Colorado Avalanche on July 28.
Ryan Dzingel was also injured in the loss at Florida; Arizona did not have an update on the center.
The Coyotes recalled goalie Ivan Prosvetov, defenseman Cam Dineen and forward Barrett Hayton from Tucson of the American Hockey League on Tuesday.
The Coyotes (0-5-1) do not have a win in their first six games for the second time in franchise history; they were also 0-5-1 in 2017-18.
"You never hope to go through it, but we're in the thick of it and there's no easy way out," forward Christian Fischer said. "It's going to take a lot of work. It's early, but it [stinks]."
Arizona will play the second of a season-high six-game road trip at the Tampa Bay Lightning on Thursday.
NHL.com independent correspondent Alain Poupart contributed to this report
