-It is strange that your new H1 effective date was on 11/16/2003. In your H1 transfer, your new employer should indicated that the h1 effective date was 11/15/2002 so that your H1 status would have no gap.
Also, your new employer sent out the H1 transfer petition might be too late (only 2-3 days before your H1 expired) and thus, it might result a gap in your H1 status because USCIS need about 10 days to put your data into their system.
So, You need to check whether the receipt date on your H1 transfer receipt was before 11/15/2002. If no, you were supposed not to work until your H1 transfer was approved, and your will have big trouble now. If yes, you have no problem and just need to provide a copy of your H1 transfer receipt and explain in the cover letter that you were qualified to work legally in the period of time from 11/16/2202---06/15/2003 based on AC21.