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
 

 Anyone please any suggestions 

You can file N number of h1b. But I am not sure if you can change job/employer during capgap extension or not. If you can't then may be you can only work with your new employer after oct 1 st

 Meraj Thank you for your suggestion. Please anyone with more information. Tried googling, no luck so far with reasonable information.