White Sox general manager Chris Getz said Tuesday on MLB Network Radio on SiriusXM that Taylor will move into the rotation next season.
Burke was charged with the loss in Game 5 of the ALDS against the Guardians on Saturday, giving up two runs on four hits with four strikeouts and no walks over five innings.
Smith didn't factor in the decision in Game 1 of the AL Wild Card Series against the Astros on Tuesday. He allowed two hits with one strikeout and no walks over three innings.
Meidroth (hamstring) will start at second base and bat eighth Tuesday in Game 1 of the American League Wild Card series in Houston.
Richards was credited with the save Sunday against Colorado, allowing one run on one hit and no walks in 3.1 innings. He struck out three.
Jones did not factor into the decision in Sunday's 4-2 win over the Tigers, allowing one hit and no walks while striking out five over four scoreless innings.
Kay (10-9) earned the win Sunday against the Rockies, striking out four while allowing one run on five hits and two walks in five innings.
Griffin went 2-for-4 with a home run and two RBI in Sunday's 4-2 win over the Tigers.
Montgomery earned the save in Sunday's 4-2 win over the Tigers, striking out two in a perfect ninth inning.