Smith, 20, returned to Bramall Lane earlier this month following a successful spell with AFC Barrow, where he scored 10 goals in 23 starts and will continue his development at Rovers.
The young forward is ineligible for Rovers’ home game with Rochdale on New Year’s Day and the FA Cup game at Preston on Sunday.
Speaking about Smith recently, Blades chief Chris Wilder said: “Tyler’s situation is that we wanted him back over Christmas to have a look at him.
“But we always knew he was going to go back out to a League One club. That’s great for him and his development.
“It’s been interesting to see him develop and come on, to see his attitude change, to see his confidence come out. He’s come out of his shell a lot more now.”