You can seek entry based on valid H-1B visa and new company's approval notice.
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You only need a new stamp if your current one has expired or is about to expire.
J Charles Ferrari Eng & Nishimura 213.622.2255 The statement above is general in nature and does not constitute legal advice, as not all the facts are known. You should retain an attorney to review all the facts specific to your case in order to receive advise specific to your case. The statement above does not create an attorney/client relationship. Answers on Avvo can only be general ones, as specific answers would require knowledge of all the facts. As such, they may or may not apply to the question.
Yes, you can enter with your H1 as long as it has not expired. You only need to get a new stamp for the H4 when the H1 visa on your passport expires.
This is only a short advice- you should not file anything without getting the full facts from a lawyer.