Yes, you may file for an extension of your H-1B while you are processing your PERM application and even through your application for adjustment of status. H1B status may even be extended beyond the 6th year (including time spent in L status) if the PERM was filed at least one year prior to the date the 6th year is reached and/or if the I-140 has been approved.,
You are entitled to 6 years in H-1B status after which you must have either filed for Labor Certification or departed the US for the requisite period. Since you re-entered the U.S. on L-1B, under "automatic visa revalidation" your L-1B was reinstated and your H-1B cancelled. You may want to extend your H-1B to reach the 6 years and file extensions after 6 years under AC21 (with a pending PERM).
Please consult an immigration attorney. This does not constitute legal advice.
To extend an H-1B beyond the sixth year, you must have a PERM/I-140 that has been pending for over one year or have an approved I-140.
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.