This might help:

"I stopped claiming because I ran out of my original EUC benefits. Do I need to send in a new EUC application or reopen my claim to get these additional EUC benefits?

No. If you already applied for EUC, no new application is required. The additional EUC amount will be automatically added to your EUC benefits.

If you stopped claiming, you will need to reopen your claim online or by calling the telecenter."


http://www.esd.wa.gov/uibenefits/fil...mpensation.php

It appears you can reopen a claim. This is for the state of Washington, but should apply to other states since it's EUC. Paid for by the federal goverment.