Become friends with the club matrons.
But seriously, the closest you can get to free access is the Saks World Elite Mastercard, which has no annual fee and gets you Priority Pass membership, but you still have to pay $27 or so per visit. Delta also offers a 30-day temporary membership for $90, which, while not free, is a great deal, especially when traveling a lot in a short period of time or with family, who can join you in the lounge for free.