I was recently able to purchase a one-day pass (not an Amex single entry) upon arrival at a Sky Club with a valid ticket for same-day travel on a Delta flight (in economy, no less!).
It was $50. No hassles. No questions. Quick and easy.
Edit: I was never able to find a way to do this through the Delta app, despite their insistence about this point. Did not try online or at a kiosk, just walked straight up to the Dragon's Den, smiled, and asked for a one-day pass.