You can also apply for asylum from any point in the US.
What do you mean by any point?
People With Visas or Other Entry Documents Can Wait Until They're in the U.S. to Apply for Asylum
If the border inspections officer allows you into the U.S. on a visa or other entry document (without your having requested asylum yet), then the way to apply for asylum is to mail in an application, with supporting documentation,
on Form I-589. You'll have a year from your entry date to do this. The form and instructions are available for free download on the USCIS website (
www.uscis.gov). Also see
Asylum or Refugee Status: How to Apply.
However, if at all possible, you should
get help from an attorney. Applying for asylum is a complex process. You must prove not only your fear of what might happen to you upon returning to your home country, but that you were persecuted based on one of five recognized grounds (race, religion, political opinion, nationality, or particular social group), and that you yourself are not barred from the U.S. due to past crimes, persecution of others, resettlement in a third country, and so forth.
A good asylum application is accompanied by things like news clippings showing conditions in the home country, medical or psychiatric reports, affidavits from witnesses or experts, and more. An attorney can help with all that (and you may be able to find one who volunteers some or all of his or her time.)
Can You Request Asylum At U.S. Border or Entry Point? - AllLaw.com