I believe you can extend you H-1B if you are eligible. Once you get you receipt, you can start after expiration. You should take into consideratin of the turn aound time for the paper work inside your company. As far as I know, it can't be faster than 3 weeks. Details depend on your specific situation, please refer to pjiang's blog:
I-129: http://blog.wenxuecity.com/blogview.php?date=200709&postID=27583
H-1B beyond 6 yrs: http://blog.wenxuecity.com/blogview.php?date=200708&postID=40272