A second-round 69, three-under-par, gave the 30-year-old former US Walker Cup player yet another chance of a maiden European Tour victory as he moved to a five-under-par 139, within a stroke of Smyth, who was an afternoon starter.
May has made four top-10 places in this year's six starts and his best finish was joint second in the Malaysian Open three tournaments ago.
He made his latest bid after again coming back from a poor start when he bogeyed the first hole, his putter helping him up the leaderboard as he holed three putts around the 25-foot mark. "It's been a good start to the season," said May. "I'm hoping to keep it going, obviously looking to go one better than my Kuala Lumpur finish.
"I've been battling a cold ever since I came here and I've twice had to fight bad starts. In the first round I started bogey, double-bogey and thought it might not be my week."
Spain's Pedro Linhart made an upward move, too, to lie second in the clubhouse on 140 after a 69, but Frenchman Jean Van de Velde lost his overnight tied second spot through a 73 for 141.
Britain's Roger Chapman produced the best score recorded at the Penina course but his seven-under-par 65 will not count for the record because preferred lies are in operation.
The 39-year-old Englishman put his round down to no longer worrying about golf because of health problems suffered by his youngest son Thomas.
"My eight-year-old son has been diagnosed as suffering with hip joint problems and told he's got to give up sport for five years when he's football- mad," said Chapman. "That brings it home to you that there are more important things in life than golf and I guess I played better by not worrying about the game any more."
His 18-year-old compatriot, Justin Rose, is worried about his game, however - especially his poor concentration.
He pinpointed his lapses in concentration as the chief reason for almost certainly missing his 15th consecutive cut since turning professional in a blaze of publicity last July after finishing fourth in the British Open. Rose slumped to a 77 and 148.
An early double-bogey rocked Rose and he virtually lost his chance of qualification with bogeys at the last two holes.