At 9600 baud it will take 1/960 seconds to transmit one byte, namely: 1.041 mS. The transmission is done by an interrupt after it is placed in the buffer, so it will commence after the Serial.write().
Clearly you need to allow over 1 mS for that to finish, however I would have thought that 2 mS would be adequate. Maybe 3 just to be safe.