OPT/H1B transisition and the gap period - work authorization?


I am about to be in the transition period between OPT and H1B and have a few questions. I am currently working on OPT at an employer that is not sponsoring H1B visas. My OPT period ends May 31st, 2014 and I think the earliest employment start date for a 2015 H1B visa is October 1st, 2014. I probably need to take advantage of the OPT H1B gap period between June 1st and September 30st, 2014, the period between my OPT end date and H1B start date. Given I have a new job offer by April 1st and my future employer will apply for my H1B on April 1st will I be able to work for either employer until the beginning of my H1B work authorization (October 1st, 2014) or am I required to stay with either employer?

For example, can my future employer hand in the H1B application while I am still working at my current firm or am I required to be with the new firm when they apply for my H1B visa? My current plan is to have a job offer by April 1st and let the new employer apply for my H1B on April 1st but stay with my current company at least until the end of May, 2014. But I am not sure if this would be possible under H1B/OPT rules.


Thank you!