The National Park Service recommends purchasing tickets prior to the day of your visit.
statuecruises.com, "the official ferry service provider" (per the above URL), says that they have ferries that depart for the Statue of Liberty as late as 3:30pm on Dec 15 and that depart from the Statue of Liberty as late as 5pm, which I think is plenty of time to explore the island to my satisfaction.
However, when I try to purchase a ticket (New York Reserve) the latest that they sell tickets for is 1pm.
So what if I want to take the 3:30pm ferry and just walk around the island? Is it not possible to purchase tickets online or in advance for that?