The full-back made his Spurs debut on the final day of last season, coming on as a substitute in the home game against Fulham.
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The 22-year-old enjoyed successful spells at MK Dons and Leeds United last season and moves to another Championship club in the form of Millwall.
The Lions' manager Kenny Jackett has welcomed the arrival of Smith, as he is missing key defensive options with captain Paul Robinson, Jack Smith, Karleigh Osborne and Alan Dunne all unavailable ahead of Saturday's trip to Nottingham Forest.
"Adam is a player I have been following for some time," Jackett told reporters.
"He's a very promising young full-back and with both Jack [Smith] and Alan [Dunne] unavailable for Saturday this was an opportunity to bring Adam in to fill that gap in the short-term and hopefully for him to get more first team experience over the next couple of months with us."