Travel on old visa stamp(valid) with a new I-797

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I was in the US on L1-B from Jan 2011 to Oct-2014. I had a COS H1-B application filed from Consulting firm A which was approved for 2014 Cap. I worked for a couple of months and left US in Dec 2014. 

That leaves about 2.2 years of H1 time that could used.

Consulting firm B filed for my H1 in April 2020 which was approved till Aug 2022. As there was a ban because of CoVid, I could only go for a stamping in April 2021. My Visa was stamped from April 2021 till Aug 2022.

 I got a FTE offer and from a new employer C, and they have started the paperwork. If the petition is approved, do I have to reappear for Visa Interview, or, can I travel with my current Visa stamping and a new I-797? I haven't travelled on the new stamp so far. The stamping has Consulting Firm B's name on it.

Appreciate your time.

Thank you!


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