If I don't fly back on September 28, 2013 will my current company notify immigration that airline tickets have closed?

Question

Right now I am employed through Company L1 with a work visa that is valid through October 2014. I have another H1B work visa through another employer that is valid through October 2013. The company I work for now wants me to go to India on a flight for September 28, 2013. I am currently on vacation until October 07, 2013 and I am looking for another job in the United States and would like to stay in the United States for one more week without letting my current company know.

Answer 1

If you have an approved H1B status then you will be able to start working for the H1 employer starting October 1st, 2013.

Answer 2

If you have an approved H1B status then you can start working with your H1 employer beginning October 1st, 2013.