Changing Employer from Cap-Exempt to Cap-Subject

My current employer is planning lay-offs in coming months. I work for this employer on cap-exempt H1B.
I am getting a job offer from a company (for-profit), which wants me to start working for them this month.

So I have to change my visa from cap-exempt to cap-subject? Do I have to wait till October to start working with new (for-profit) employer?

Is it possible for me to transfer the visa from cap-exempt to cap-subject and start work before october?
Does such process have any kind of premium processing ?

Thanks

Before you can work for an employer that is subject to the cap, you must be counted against the cap. That hasn't happened yet. Based upon what you've said, you can apply starting on April 1 to start working for that employer in October.