silence43,

You can apply with your current I-20 but as you'll have the gap, USCIS will approve your H1 but not the I-94. If you apply with a new I-20 now that covers the gap, you'll get your I-94 also and will be in H1 status starting Oct 1.

If you apply with your current I-20, USCIS won't issue your I-94.

To get your I-94, you either a) Need to go out of US before your OPT grace period expires, apply for H1 stamp and come back in the last week of September OR b) Join ESL, Kaplan or a new school in July (before OPT grace period expires) and file H1 amendment petition with the new I-20. The amendment petition can be filed only after your original H1 petition is approved.

If you are applying with your old I-20, file under Premium processing so that you get the H1 decision and get enough time to transfer before grace period expires.