After you get job offer, the company can file a H1 for you with the copy of the H4 receipt or H4 approval notice. Depended on how long the gap of H1B is, you have two options
1) If the gap is within the two months, you are allowed to do so-call H1B transfer, jumping on work right after receving the receipt according to a CIS memo back to the legacy INS.
2) If the gap is larger than two months, you may need to wait until the new H1B is approved to start work.