The 40-year-old 14-time major champion, who twice had surgery during 15 months out, hit seven birdies as he moved into joint ninth on six under par.
From trailing by nine after a 73 on Thursday, he is now six off the lead.
"Yesterday I had it going and lost it. Today was different," Woods said.
He told the Golf Channel: "It feels good. I played the middle of the round better, kept the momentum going and moved myself up the board."
First-round leader JB Holmes shot a one-over 73 to slip to joint seventh as his fellow American Dustin Johnson moved into a share of the lead with Hideki Matsuyama of Japan.
Matt Kuchar and Louis Oosthuizen - who both shot 67s - are tied for third, two shots back, while Bubba Watson had a nine-under-par 63 to move three shots off the lead.
Behind him, Brandt Snedeker had an eight-under-par 64 to move one shot ahead of Justin Spieth and Holmes on eight under.
Olympic champion Justin Rose, playing his first event after a seven-week break, withdrew with a bad back before play got under way at the four-round event in the Bahamas.
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Tiger Woods recovered from a mixed first round by shooting a brilliant seven-under-par 65 as he continued his return to competitive golf on day two of the Hero World Challenge.