No, you cannot legally remain in the US after the expiration of your I-94 in September. You'll need to take a 3 week trip somewhere outside the US.
You can also consular process in Canada or Mexico and come back. You can be admitted up to 10 days prior to October 1 on an H-1B visa.
Behar Intl. Counsel 619.234.5962 Kindly be advised that the answer above is only general in nature cannot be construed as legal advice, given that not enough facts are known. It is your responsibility to retain a lawyer to analyze the facts specific to your particular situation in order to give you specific advice. Specific answers will require cognizance of all pertinent facts about your case. Any answers offered on Avvo are of a general nature only, and are not meant to create an attorney-client relationship.
You did not mention your company's prospect of extending your L-1B status. If you are currently maintaining L-1B status and the company does not wish to terminate you, they can seek an extension of your status.
This response is general in nature and cannot be construed as legal advice, given that not enough facts are known. Any comments offered on Avvo are of a general nature only, and are not meant to create an attorney-client relationship. If you would like additional information based on this response, please contact my office at 510 657 7665 to schedule a consultation.