I asked Jon at Encanto Farms to recommend five good closed-eye alternatives to my souring-prone Brown Turkey, and he suggested Celeste, Black Mission, Vista, Panache, and Black Madiera, and as backups added White Adriatic, St Jerome, GM1, Strawberry Verte, Adriatic, Portland 4, 135-15s, Black Weeping, Galbun, LSU Gold, and LSU Purple.
I also emailed "Just Fruits and Exotics Nursery" http://www.justfruitsandexotics.com/Figs.htm and asked which of their figs were closed-eye varieties, and they emailed the following (I'll skip duplicates from Jon's list):
Alma--moderately closed, small eye--honeydew closes eye
Giant Celeste (Tiger) partially closed
Golden Celeste (Champagne) partially closed
LSU Improved Celeste (O'Rourke) partially closed
Conadria--small, tight eye
Excel--medium, tight eye
Green Ischia--closed
Hunt--closed
Tena--medium, tight eye
Texas Everbearing--moderately closed, small eye
I've had no other dealings with Just Fruits beyond this email exchange and can't vouch for the accuracy of their information, but they seemed friendly and responsive.