You can change employers but because the H-1B is an employer specific visa, in order for you to start working for another employer, that employer needs to file a H-1B visa petition on your behalf.
Your H-1B classification is still good as long as you are still employed by the Petitioner and you are abiding by the terms of the H-1B. Even though the visa expires, the status does not. The status is good as long as the I-94 is valid. As long as you are in status in the U.S. you can switch H-1B employers. You can use portability or you can process the case (by normal methods or by premium processing) and move to a new H-1B employer.
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