Either transfer to the startup in H-1B or O-1 status or file a part-time concurrent H-1B through the startup and continue to work for your current employer unti

Either transfer to the startup in H-1B or O-1 status or file a part-time concurrent H-1B through the startup and continue to work for your current employer until the part-time concurrent H-1B is approved and then make the complete switch. (This is a more conservative approach.)