Individuals in J-1 status must submit any requests for an extension of their program, a transfer to another school, a change of level or Academic Training before the current DS-2019 expires.

When an exchange visitor completes his or her program as scheduled, the exchange visitor's SEVIS record status will automatically change from "Active" to "Inactive," the day after the program end date. 

Once a student's program is over, they are in their 30-day grace period. During this 30-day grace period, participants are no longer in J-visa status, and are under the jurisdiction of the USCIS. The USCIS grants this period to allow participants to settle their affairs and to prepare to return to their home countries. Program participants may no longer continue and/or complete exchange activities, nor may they work. Although participants may travel in the United States, it is recommended that they do not travel beyond the borders of the United States as they may not be permitted reentry.

If you decide that you would  like to apply for a change of status, you should do so before your DS-2019 expires.

"Many advisers and attorneys recommend that a change of status application be filed within an individual's program period as reflected in SEVIS and on the individual's Form I-20 (F status) or DS-2019 (J status), and that filing during a grace period continues to be risky. There have been reports, for example, that USCIS will also look to the beneficiary's SEVIS record status when determining whether an individual was in valid nonimmigrant status at the time the change of status application was filed. This is particularly problematic for J exchange visitors in their 30-day grace period, given DOS's narrow interpretation of the J 30-day grace period, and since SEVIS automatically changes J SEVIS records to Inactive the day following the program end date."