You mention the NPS website. Also on that same page, it states:
Due to its popularity, access to the pedestal is limited and requires
reservations. Statue Cruises is the official source for pedestal
reservations as well as the company that provides ferry transportation
to the island. Advanced reservations may be made online or by
telephone.
So that may be a far easier way of ensuring you get them.
How early is early enough in the morning is going to depend on the season, weather, events and luck. However, the first ferry with the official Statue Cruises is 9.30 in the morning from Battery Park, so make sure you’re there before then. One would figure it’d be easy to pass there earlier in the morning, see if there’s a queue, join it if there is, or go get a coffee and come back, if you’re really certain about getting there early.
Credit:stackoverflow.com‘