The 64-year-old's 7-under-par round left him at 9-under 135 through 36 holes -- two better than last week's winner, Willie Wood.
Jenkins is looking to break the record held by Mike Fetchick, who was 63 when he won the 1985 Hilton Head Seniors Invitational.
Wood, coming off his victory at the Dick's Sporting Goods Open, carded a 4-under 68 to claim sole possession of second at minus-7.
First-round leader and defending champ Mark Calcavecchia slogged to a 1-over 73, falling into a third-place tie with David Eger (68), Mike Reid (68), Jay Don Blake (70) and Mark O'Meara (64) at 6-under 138.
Celebrity Families of 2014 [PHOTOS]