Can I transfer from H3 to H1B ?

I am currently holding H3 visa which is effective from this september and will expire spetember 2015, and I am working at company A with that H3 visa. If I plan to jump over to company B next year before 2015 H1B application, and company B is willing to sponsor me to apply for 2015 H1B.
My question is: can I start working for company B immediatly after approval of 2015 H1B application, or I can only start working for company B on 2015 October when 2015 H1B is effective ?