As the breeze picked the players struggled around Clearwater and only Australian Heath Reed shot a red number of those who teed off in the afternoon.
Hancock said he didn't come to the Open with expectations of shooting low.
"I would have taken probably four even pars or four one-unders. It's a pretty tough golf course, the rough is pretty thick. Four under is really good,'' he said after his round.
"Anything could happen tomorrow. Who knows what the weather could do - it could be wet tomorrow. Nobody knows in New Zealand, do they?''
The weather is forecast to be fine tomorrow but the wind is likely to pick up again in the afternoon, which makes finding birdies tough on the greens.
The fact that very few people would know who Hancock is wasn't lost on him and he said the Kiwi contingent would be chasing him during the final three rounds.
"I sneak under the radar, for sure. Don't worry about the Kiwis, they'll come home pretty hard. There's some good players here.''
Of the Kiwi challengers, rookie pro Fox signed for the best score as the Aucklander carded a two-under 70 to be right in the mix.
Fox, who began on the 10th hole, was five-under through 11 before he lost his ball on the third hole and dropped two shots, before he gave up another on the par-four sixth.
"My first 11 holes were really, really solid and then when the wind got up on the front nine I just sort of struggled to get myself decent looks at birdie. But my short game held up. I hit a lot of shots close today, which helped,'' Fox said.
The 25-year-old will tee off at 12.20pm today but said he was still confident he could pull back Hancock even if the wind got up.
"There's still lots of birdie opportunities out there as long as you can sort of get through probably 18, seven, eight and nine unscathed there's still a score there.''
Reed, who defied his world ranking of 1479 equal to shoot a one-under 71, helped his cause with an eagle on the par five 14th when he holed out from 85 metres with his sand wedge.
One-under was the fashionable score of the day as nine players banked a 71, including Kiwis Vaughan McCall, Gareth Paddison and Brad Shilton.
Only 12 players went under the card and New Zealanders, Mahal Pearce, Mark Brown, Mathew Perry and Pieter Zwart all shot an even-par 72 but they had the luxury of a morning tee time.
Former European Tour pro Greg Turner, who is playing his first Open since 2002, logged a two-over 74 that was good for a share of 33rd place.
Michael Hendry came in to the event as one of the joint-favourites but the Kiwi pro shot a two-over 74, while former web.com tour player Josh Geary carded an eight-over 80.
