The 42-year-old has displayed patience, outwardly at least, knowing that even a player of his calibre needs some time to get back to near his best after missing so much time through injury.
He rebounded from an early double bogey to storm to the clubhouse with a six-under-par 66, including an eagle on the 16th.
A week that began with McIlroy full of confidence ended in despair as the 29-year-old paid the price for too many mistakes off the tee at TPC Sawgrass, where he closed 16 shots behind runaway leader Webb Simpson on one over par.
Simpson was the seventh player to shoot 63, the last five since 2013.
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"You start just kind of laughing", said Simpson, whose halfway score of 15-under matched the tournament record set by Day two years ago.
However, with a five-shot lead Simpson does surpass Day (four strokes) for the largest 36-hole lead in Players history.
Simpson then two-putted for par at the par-4 18th.
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Charl Schwartzel (66), Patrick Cantlay (68) and Danny Lee (66) are Simpson's nearest rivals heading into the penultimate round.
On the 532-yard par-5 second hole, Kiradech Aphibarnrat reached the green in 3 and sunk a 3-foot putt for birdie.
The Northern Irishman struggled off the tee, hitting only 6 of 14 fairways and found 10 of 18 greens in regulation.
But he pulled a driving iron off the tee at the 10th into sand and dropped another shot, although he got it back when he drove the green at the 12th only to block his tee shot to the right of the 13th green and card his fourth bogey of the day.
A slight mis-hit and his tee shot to 17 stalled in the air, it just pitched on the front bank next to the bunker, but rolled back into the water.
He added: "I wish I could repeat it more often but, honestly, it was just [about] a better start".
"I think I started tidying up a few of the areas I had (trouble with) last (week) and I think the swing is coming along".
Woods - a 14-time major champion and two-time victor of the PGA Tour tournament - hit a 71 to reach one under through 36 holes, a shot above the cut line.