With a pedastal ticket, can I start the ferry at liberty state park(NJ side), and use the same ticket to take the ferry to go to the battery park(NY side)? Thanks in advance.

Yes, you can.

Yes

Thank you for reply. I would like to get more details. I saw on the statue of liberty website that,the ferry from NY side goes to the liberty island and then to the ellis island and back to the NY side. But the ferry from NJ side goes to the ellis island first and then liberty island and then back to NJ side. So if I start at NJ side, and take the ferry to ellis island and then to liberty island, can i go back to ellis island and take ferry to NY side?