In Friday's modified alternate shot format, Australian Day and American Tringale produced an eight-under-par 64 to reach 25-under overall at Tiburon Golf Course in Naples, Florida.
Day said he feels the pressure will be on in the final round, with each player playing their own ball on every hole.
"I would expect a lot of guys, even if your partner is in play, because it's always better to have two balls in play than just one," said Day, as he and Tringale built on their two-shot overnight lead.
"I would rather have two opportunities at birdies than just one."
Northern Ireland's McDowell and American Woodland made their move in the unique PGA Tour tournament, shooting a 63 to be 22 under and in the mix heading to Saturday's final round.
McDowell said he feels he and Woodland are playing well enough to reel in the leaders.
"Two balls in play, you know, it means no one can hide," he said.
"That was kind of why I really feel like myself and Gary have a good shot tomorrow because we're both really playing well tee to green."
Defending champions Harris English and Matt Kuchar posted a 66, to sit four shots off the pace in outright third.
Charles Howell III and Scott Verplank shot a 64 to be 19 under and fourth, ahead of three pairs tied for fifth a shot further back.
Keegan Bradley and Camilo Villegas (67), Billy Horschel and Ian Poulter (65) and Patrick Reed and Brandt Snedeker (65) are all square at 18 under.
Saturday's third and final round will see the 12 pairs play under the better-ball format in the final round of the $3.1million prize pool.