Smith was given a guard of honour by both sides before the first round clash and received a standing ovation when he was substituted early in the second half in his 273rd and final appearance for the club.
However, Premier Division side Mickleover proved more clinical in front of goal than the Daniels to ensure a losing finale for the popular player.
The Derbyshire visitors took the lead midway through the first half through Nathan Jarman and then doubled their advantage on the stroke of half-time through Isai Marsella.
Marsella then netted his second and Mickleover's third with six minutes remaining to secure their success.
The Daniels had made a number of changes to their starting line-up with goalkeeper Joe Tarris and right-back Joe Bates making their first starts while youngster Harry Peasgood was handed his full debut.
Manager Graham Drury commented: "We made seven changes from Saturday and it was nice to give Harry his full debut and give Jordan a send-off.
"Mickleover made changes as well and it was good competitive game.
"I thought we created a lot, but there was no end product. They got their chances and stuck them away.
"They were a bit fortunate with their first two goals which were both richochets into the paths of players.
"But we found ourselves 2-0 down after again having quite a few opportunities in the first half to pull the trigger. We're not ruthless enough at the minute though which is perhaps hurting us.
"We need to get back to scoring goals and taking our chances when they come."
Stamford could have opened the scoring in the first minute when debutant Bates fizzed over a dangerous cross to the back post which Sam Grouse bundled into the side-netting.
The experienced Jarman showed Stamford had it should be done when he broke the deadlock after 24 minutes after finding space at the back post to beat Tarris with a composed finish.
There was an immediate response from the Daniels with Eliot Putman curling an effort against the Mickleover crossbar before seeing another
well-struck effort safely gathered by Sports custodian Lewis King.
Mickleover player-boss John McGrath then cracked a sweet free-kick against the home crossbar with Stamford replying as Stan Mugisha flashed a shot narrowly wide before teeing up Grouse who failed to connect with the pass.
Rob Morgan saw a dipping drive tipped around the post by King before Marsella sneaked a effort past Tarris only for Peasgood and Bates to join forces and scramble the ball off the line.
The visitors doubled their advantage on the stroke of half-time when the ball broke to Marsella in the box and he fired past Tarris.
Stamford pushed to reduce the arrears after the break with Putman firing another effort over before Liam Adams latched onto a fine pass from Peasgood only to be denied by a top stop from King with Grouse firing the rebound side.
Mickleover then sealed their passage into the next round with six minutes remaining as Marsella prodded home from close range following a corner.