The American is yet to win on the PGA Tour in 365 career starts but gave himself a decent chance of ending his barren run with a three-under-par 67 at the Sea Island course in Georgia.
That took him to 10 over par overall and moved him alongside compatriot Chris Kirk at the top of the leaderboard.
Baird birdied the third and seventh holes but was back to level par for his round after back-to-back birdies at the ninth and 10th.
A run of pars followed but the 41-year-old thrust himself into contention ahead of Sunday's final round with a run of three birdies from the 16th.
Kirk stormed to life in his final six holes to move to 10 under.
The overnight leader was two over for his round after bogeying the par-three 12th, but bounced back with three birdies in his next three holes before picking up another shot at the last.
Kevin Stadler had the round of the day with a fine 65 which helped him move one shot behind the leaders.
The American sits alongside compatriot Brian Gay and Australian John Senden.
Brendon Todd and Scott Brown are well placed to make a move on Sunday on eight under, while fellow American Jason Kokrak is a shot further back, and Matt Kuchar and Webb Simpson were among the four players four shots off the lead.