You should be able to work as

soon as you get transfer receipt. So the most important thing for you to do is to push the school to submit your transfer IMMEDIATELY. Between your old H1 expiration date and new H1 transfer receipt date. Your status is illegal. Theoretically you have to leave the US immediately, but in practice a 10 day grace period is usually allowed for you to prepare for the leave. So you have to make full use of the rest time.