Australia's David Warner, who struck scores of 154 and 335* against Pakistan, gained 12 positions to take the fifth place.
He has 764 rating points and went past Ajinkya Rahane, who dropped to sixth (759).
England captain Joe Root, who scored a double ton in the second Test against New Zealand, rose to seventh.
Meanwhile, an in-form Marnus Labuschagne, has taken the eighth position.
Smith will hope to regain the top spot shortly
Australia will be up against New Zealand in a three-match Test series, starting December 12. Smith will be eyeing to score valuable runs against Kane Williamson's men. That could see him get past Kohli in the rankings. India's next Test assignment is in 2020.
Pat Cummins continues to rule the show in bowling
In the ICC Test Rankings for Bowlers, Australia's Pat Cummins continues to occupy the first position.
He is the only bowler to have taken 50-plus Test wickets in 2019.
West Indies skipper Jason Holder, who shined against Afghanistan, moved up to third.
Josh Hazlewood gained one place and is eighth.
India's Mohammed Shami gained one position to enter the top 10.
Starc moves up to 14th in the rankings
Australia's Mitchell Starc was the biggest gainer. The left-arm pacer moved up to 14th in the rankings and has 752 points. He is behind Tim Southee. Meanwhile, Umesh Yadav and Nathan Lyon, who claimed respective fifers recently, gained one place each.