Alker, whose highlight of the season came when he finished tied 19th at the Open Championship at Royal Lytham and St Annes, made seven birdies in all - including four in a row on his back nine - and two bogeys as he turned his fortunes around.
Lee and Alker need to finish inside the top 19 players plus ties to advance to the final stage.
Meanwhile, Manawatu professional Tim Wilkinson has kept pace in his attempt to return to the PGA Tour at the second stage at the Hombre Golf Club at Panama City in Florida.
Wilkinson, who carded a four-under par 67 to be tied for ninth place after round one, backed that up with a one-under par 70 to be in a share of 13th place at the halfway stage.
He is on a five-under par total seven shots back from American Scott Sterling (65, 65) who leads the event by one shot.
Wilkinson, who finished 57th on the web.com Tour order of merit, needs to finish in the top 19 and ties to advance to the final stage.
The news wasn't as positive for Tauranga professional Josh Geary at the second stage event in Texas.
The 28-year-old, who had carded two rounds of 70 to be tied 27th at the halfway stage, lost his way in the third round with a four-over par 76 at the TPC Craig Ranch to drop to a share of 50th place.
Geary, who has played only five events on the web.com Tour in 2012, is on an even-par total 17 shots back from the leader Matt Jones.
The two-time winner on the Charles Tour needs a great round tomorrow to finish inside the top 20 players and ties to safely progress to the final stage.
Finally Aussie Peter Fowler, a golfer many on this side of the ditch consider an "honorary Kiwi" as he lives in Auckland, has a chance of qualifying for the Champions Tour in the States.
The 1993 New Zealand Open Champion, who won the Senior European Tour Order of Merit in 2012, has fired a three-under par 69 in round three at the TPC Eagle Trace in Coral Springs Florida to be in a share of ninth place.
Fowler is seven shots back from the leader Gene Jones and needs a good final round to advance to the Champions Tour.
He needs to finish in the top five to earn fully-exempt status through the re-order in the 2013 season and if he finishes in the next seven players he will be conditionally exempt.