My situation is very similar to the one described at http://www.murthy.com/news/n_h1bemp.html
I am working in H1B status for a university, a cap-exempt employer. The current H1B and I94 are valid till June 30, 2012. I obtained a new job offer from an employer, which is cap subject. The new employer has submitted a LCA requesting start date from April 1, 2012 and will file an H1B petition on April 1, 2012, requesting a start date of October 1, 2012. October 1 is the earliest possible start date, because when one moves from a cap-exempt university employment to a cap-subject employment, s/he must be counted against the H1B cap.
An experienced lawyer working for the new company assured me that I work for the new job between April 1 and October 1 2012. The murthy.com also explains that in the post I mentioned above.
But I still have a confusion because my current I94 will expire on June 30, 2012 and I believe that the new I94, if the H1B from the new employer is approved, will start from Oct 1, 2012.
Can I still work and stay legally in the US from July 1 to Sept 30, 2012?