Vaughn Taylor, who had missed his last two cuts since winning the Knoxville Open in June, fired a 65 to finish three shots off the lead at 15-under-par 201. Taylor was joined by Tommy Tolles, Jason Bohn and Rob Bradley in a tie for fourth.
Palmer started the day in a tie for fourth with Thompson and Carter at Black Creek Club and picked up two birdies and an eagle over his first six holes.
The 26-year-old bogeyed the ninth to make the turn at 14-under. Palmer, who won the Clearwater Classic earlier this season, tallied four birdies on the inward half for his share of first.
'A couple of years ago I would have said that the harder the golf course the better I'll play,' said Palmer. 'I'm not one to get into shooting 20- to 25- under to win but last year I played a mini tour event and shot 26-under. It made me believe that I can do it.'
Thompson, who is seeking his first career victory on the Nationwide Tour, ran off three consecutive birdies starting at the par-5 fourth to move to 14-under.
The 24-year-old birdied the 11th but found trouble with a bogey at the par-4 12th. Thompson responded well with three straight birdies from the 13th before dropping another shot at the par-4 16th.
Thompson countered again and birdied each of the last two holes to join Palmer in first.
'You've got to work the ball both ways on this course and pay attention to where the trouble is,' said Thompson. 'There is a lot of risk-reward on these holes but I've been playing pretty conservatively and letting my wedge game do the work this week.'
Jimmy Walker, Franklin Langham and Steve Haskins each fired rounds of 64 on Saturday. They joined Matt Peterson and Ken Duke in a tie for eighth at 14-under-par 202.
Kevin Johnson was one shot further back at 13-under-par 203.
Bryce Molder, Omar Uresti, Jason Dufner, Jeff Freeman and Ted Purdy finished tied at 12-under-par 204.