Well, #*@(%&(@. I just spent over an hour crafting a reply, and it got eaten.
Short version: You have to have advance tickets to the crown of the Statue of Liberty, and they are sold out until August. You can get a pedestal ticket at 2:00 or 3:00 on the day you'll be there, and it would include Ellis Island.
https://www.nps.gov/stli/planyourvis...-the-crown.htm
Maybe some tours companies buy up crown tickets and sell them only with their tour--I have no idea, but you have to have a name on each ticket, so I don't know how they'd do it, but you're talking about New York City, where the impossible happens all the time.
Liberty Harbor is fantastic if you're visiting NYC on your own, or if you're there for several days. Since you're there for just one day, you might consider Newburgh KOA, which is half the price of Liberty Harbor and has full hookups instead of electric/water, and has a tour that leaves from the campground, and it includes the Statue of Liberty (but I assume not going into it). A very hassle-free way to do a one-day visit, if you don't need to schedule for a trip to the crown of the Statue of Liberty.
There are a million tours of NYC. I found one that involves 5 miles of walking over 6 hours:
https://www.nycgo.com/tours/half-day...all-group-tour
You ride subways and take the Staten Island Ferry (not the same as the ferry that goes to the Statue of Liberty), and ends by about 4:00, so you could go to the 9/11 Museum (last entry at 6:00). Or you could skip that and use your knowledge of the subway system to go to places like Greenwich Village, Chinatown, Central Park, etc, on your own. And Times Square at night.
There are other tours that use subways and probably don't involve as much walking. Or you can take a "regular" tour; I liked the one I took some friends on on a double-decker bus with an open top.
That's the very short version. I can't believe all that I wrote went away. It was really good.