The 36-year-old former US Open champion has fallen to 85th in the world rankings after only one top-10 finish in 25 events this year.
But he had nine birdies to reach 12 under, one ahead of Derek Fathauer.
Scot Russell Knox, who sealed his first Tour win in Shanghai last week, had six birdies in a 65 and is five shots back.
It is an impressive recovery from McDowell, who made a double bogey at the first hole on Thursday after sending his opening tee shot out of bounds and was within a fraction of doing the same with his next attempt.
"I hit my second ball further right than I hit my first - I figured I could be going home very, very soon. Thankfully, my second ball kicked in-bounds," said the Ryder Cup player, who also bogeyed the first hole in his second round, which he began from the 10th.
McDowell birdied four successive holes at the El Camaleon Golf Club after making his solitary bogey of the day.
"I drove the ball much, much better today than yesterday and putted just as good, really seeing these greens well, putting decisively and making a few," he said.
"It's been a frustrating year but that's the game of golf we know and love and you've got to take the rough with the smooth and keep trying."
Justin Leonard, the 1997 Open champion, shares fifth after mixing six birdies with three bogeys in a 68, but Englishman Greg Owen's 70 left him at one under, which was one too many for the halfway cut.

Officials said that a suicide bomber blew up a car in the city of Dammam in Eastern Province.
The Islamic State (IS) group has said that it was behind the attack in a statement released on social media.
At least 21 people were killed in a similar incident one week ago, when a suicide bomber struck during Friday prayers at a Shia mosque.
The attack at the Imam Ali mosque in the village of al-Qadeeh was the first to be claimed by the Saudi branch of IS, which was formally established last November.
In Dammam on Friday, a suicide bomber detonated his explosive device outside the al-Anoud mosque in a busy neighbourhood.
The two suicide bombs in seven days against Shia mosques are intended to inflame sectarian tension, the same tactic that IS uses in Iraq and elsewhere.
With a Saudi air campaign against Shia rebels in Yemen and a ratcheting up of the confrontation between the Saudis and Iran across the region, the jihadists are seizing the opportunity to sow more hatred.
Hardline Saudi preachers have long denounced the Shia, and that rhetoric has only grown fiercer. The Shia have called on the authorities to stem the flow of hate speech.
King Salman has promised to crush the militants and many Saudis have been expressing their solidarity with the Shia.
IS still has a very long way to go in its aim of destabilising Saudi Arabia through its sectarian divide, but it is using the only method it knows - sectarian violence.
A witness told Reuters news agency that he had seen a "quick explosion" and that he had been told that men had prevented the attacker from getting inside the mosque.
Security forces had become suspicious of a man in a car near the mosque and had gone to investigate it. It exploded as they approached the vehicle, according to the Saudi Press Agency, SPA.
One of those killed is believed to have been the driver of the car.
In a statement published online, IS said "a soldier of the Caliphate" was behind the attack on the Shia community. It identified the bomber as Abu Jandal al-Jazrawi.
Saudi Arabia has previously been threatened by IS. It is part of a US-led coalition against IS in Syria and Iraq.
The Saudis are also leading a coalition of Arab states in an air campaign against Shia rebels in Yemen.
The country's large Shia Muslim minority is mainly based in the Eastern Province, and there have been sporadic protests by Shia there to call for more rights.
Northern Irishman Graeme McDowell carded a 63 to take a one-stroke lead at the halfway stage of the PGA Tour's OHL Classic in Mayakoba, Mexico.
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A bomb has exploded outside a Shia mosque in Saudi Arabia, reportedly killing at least four people.