You can but you have no status from Aug to Oct. nothing keeps you here. Either your employer needs to extend your L or you must leave when your L status expires and return after sept 20. Talk to company attorney or hire your own attorney. I work with many Ls and Hs who have the company attorney do the work for the company but hire me to consult on what is best for them and their career path. Remember, the company attorney works for the company- not you. 415-291-8122
The change of status portion of the petition will not be granted as your status expires in August. You will need to obtain an H-1B visa stamp in September in order to be able to enter the US and be in H-1B status.
Your employer needs to retain an experienced immigration lawyer to review all the facts, advise them, and handle the case. Your employer can find one through http://www.ailalawyer.com.
J Charles Ferrari Eng & Nishimura 213.622.2255 The statement above is general in nature and does not constitute legal advice, as not all the facts are known. You should retain an attorney to review all the facts specific to your case in order to receive advise specific to your case. The statement above does not create an attorney/client relationship. Answers on Avvo can only be general ones, as specific answers would require knowledge of all the facts. As such, they may or may not apply to the question.
I concur with my colleagues.
NYC EXPERIENCED IMMIGRATION ATTORNEYS www.myattorneyusa.com; email: firstname.lastname@example.org; Phone: (866) 456-8654; Fax: 212-964-0440; Cell: 212-202-0325. The information contained in this answer is provided for informational purposes only, and should not be construed as legal advice on any subject matter.