The way I read it on the website, it says that if the original 20 weeks run out, those who exhaust the original 20 weeks of (Tier 1) federal benefits will automatically receive a recomputation of their total benefits payable, adding up to 13 weeks of Tier 2 federal extended benefits to their balance. What does this mean?
You have exhausted all the weeks of weeks of your first extension. The 13 weeks are a second (or third) extension, not remaining weeks of the first extension.
It is supposed to happen automatically. You should not need to do anything other than continue filing claims.