Answer
Extension is possible only if the program rules permit it. The worst case scenario in extension or new 2019 as I see it can be only that you have to go get a new via stamp.
Form DS 2019 FAQs
Wednesday, September 21, 2011 - 23:41
J-1 extension beyond 5 years
Question
My J-1 at Cornell Psychology Department expired on Aug 31 2011 after 5 years. Currently I'm spending my grace period. I'm exempt from the 2 years rule. I was wondering if it could be further extended. If not, would it be possible for my department to issue a new DS-2019 (J1) instead? I have only been working here for about a year and have been at a different university before.