Should I apply for second benefit claim while still on EUC?

In Texas, my company closed May 26, 2011. I will be eligible for a second claim May 27, 2012. However, I am in the middle of Tier 2 EUC at the moment. Should I make the claim at the end of May or wait for Tier 2 and then Tier 3 to exhaust somewhere in August? Also, not sure whether this is pertinent but my Temporary Unemployment Benefits (EUC claim) began 10/16/2011, when my regular benefits expired. I worked some part-time over Christmas, but never stopped payment requests--- I just reported the income during those weeks. But again, my main question is: do I need to apply at the end of May or wait until August when EUC runs out?

I'm not really sure what you mean, but from what I understand, your BYE will end on 5/27. You should be informed of this because you will have to refile at that time.