Hello Immigration Gurus, I need a suggestion. Here is the senario.

I am working as a consultant for a staffing company (say  AAA ).
They filed my H1B on April and it got approved.
My OPT got auto extended till the end of September because my H1B was approved.

I was working with a company ( say BBB ) and my contract is up now. Currently I am looking for other position.

Here are the things I need information on.

My Current OPT ( 29 month extension already taken ) expries on 2 week of July.

My question is 

If a new company ( say CCC ) offered me a fulltime position and ready to sponser my H1B, can I file a new H1B under CCC ? Or do I have to wait till November ( after 2 pay period)  to transfer my H1B from AAA to CCC.

If there are any other ways please suggest me.

Thank you for your help