The move is retroactive to December 20, for a lower-body injury believed to be a groin issue, according to the Arizona Republic.
Smith has gone 15-10-3 with a 2.52 goals-against average and two shutouts in 28 games for the Coyotes this season.
To take Smith's place, Phoenix recalled goaltender Curtis McElhinney from the club's AHL team in Portland. McElhinney was 9-11-0 with a 2.95 GAA in 21 games with the Pirates. The 28-year-old has yet to suit up in the NHL this season.
In addition, forward Patrick O'Sullivan was recalled from Portland. The 26- year-old posted two goals and three points in 17 games with the Coyotes in 2011-12.