An employer can seek to hire you and for that request a nonimmigrant worker petition on your behalf - USCIS form I 129 and ancillary forms. You do not have any grace period on L visa unless the employer extended your employment for 30 days to give you time to leave, kept you on the payroll and paid your wages during this time. H1B for the most part, with some exemptions, is subject to CAP which means that a new H1B petition can only be filed comes April 1 and not a day before. You are not allowed to stay in the USA a day pass the employment ended day on L.
Answered on Oct 29th, 2015 at 12:34 PM