The process is no different for target, employment, fully carry, etc...
Download this form:
http://www.westchesterclerk.com/images/stories/pdf/3 1 Amendment Form.pdf
Make 2 copies, fill them both out (fill out top portion of form including your pistol permit #, fill out #3, sign the bottom of the page, disregard the second page it's only for restriction or duplicate licenses)
Mail both copies along with a $3 check/MO (made out to the "Westchester County Clerk") to:
Office of the Westchester County Clerk
Pistol Permit Unit
110 Dr. Martin Luther King Jr. Blvd.
White Plains, NY 10601
Then wait...
OR you may go to the clerks office, pick up two of the forms, full them both out and drop them off at the clerks office again, but I find it's just easier to do it all by mail.
All the details are here:
http://www.westchesterclerk.com/index.php?option=com_content&view=article&id=55&Itemid=47