No. You are not even in H-1B status yet. It is only going to be effective on October 1. You cannot seek to "transfer" what you don't yet have, I.e. H-1b status. Can only seek to "jump ship" to anther H-1B employer after October 1, once you started working for the H-1B employer and can prove it with pay stubs. Otherwise your first H-1B could be scrutinized for possible fraud.