The Magpies sacked manager John Sheridan on Monday, after a run of nine successive defeats left them one point above the League Two relegation zone.
The 36 year old has been a player-coach at Notts since joining in May 2014.
"The important thing is for everyone - players, staff and fans - to stick together," Smith told the club website.
"There's no getting away from it, it's been a very, very disappointing season so far. But there are lots of points still to play for and we are determined and confident of putting things right."