Smith, promoted to open in place of Keaton Jennings, scored his first hundred in 25 first-class innings at Trent Bridge and closed on an unbeaten 119 off 308 balls to lead the visitors to 297 for six - 23 runs adrift of Nottinghamshire's first-innings score.
Smith's century arrived during his 12th consecutive hour on the field, having faced 270 deliveries. On a surface that offered him little assistance, England bowler Swann bowled 20 tidy but wicketless overs on his return to competitive action.
Yorkshire hit back strongly after Derbyshire opener Chesney Hughes amassed 270 not out on the second day at Headingley.
The 22-year-old Anguilla-born left-hander guided Derbyshire to 475 all out and was only five runs short of his county's highest individual score, set in 1896 by George Davidson, when last-man Tim Groenewald was out for a duck.
Joe Root (75 not out) and Phil Jaques (15 not out) were at the crease for Yorkshire at stumps, with the home side 164 for one in reply after after Adam Lyth's earlier knock of 69.
In Division Two, England captain Alastair Cook made a gritty 59 in more than four hours while Graham Napier top-scored with 74 to guide Essex to a first-innings total of 254 at Chelmsford.
Hampshire reached 77 for two in their second innings at the close of the second day after David Masters removed the Hampshire openers cheaply before they staged a recovery, with Liam Dawson and George Bailey posting a 56-run stand. Hampshire lead by 20 runs with eight wickets remaining going into day three.
Benny Howell claimed career-best figures of five for 57 as Gloucestershire dismissed Leicestershire for 250 on the opening day at Grace Road.
At one stage Leicestershire were 52 for five, before a stand of 125 between Shiv Thakor and Matt Boyce sparked a recovery. Thakor top scored with 75 and Boyce also made a half-century. Veteran Claude Henderson then fired 33 off 30 balls to help earn Leicestershire two batting points, with Gloucestershire 31 for two in reply after losing the wickets of Chris Dent and Dan Housego.
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