I went about a month ago on a Wednesday, and had no problems at all finding a spot. Although I wouldn't wait till to late to go, cause your pretty far from civilization not to get a spot. Good luck with your trip.
The earlier you go the better. We headed up around 5am. Got there around 6 and had the pick of spots! Then watched as everyone started flooding in throughout the day as the spots started to fill up, fairly quickly throughout the morning. Earlier the better!
Got there around 9am and waited until people left. Go around the spots and check the cards usually says what day they check-out.
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