H1B Re-stamp Question

I got my first H1B in India, I have I-797B valid up to Sep 2021. The date is the same on my visa. Once I came to the US, my designation is changed and I got a new I-797A valid up to Oct 2022. My question is do I need a visa re-stamping if I want to travel after Sep 2021, or can I travel with my existing visa and new I-797A until Oct 2022?

If I need a re-stamping, can it be done only outside the US (India, Canada, or Mexico)?

For reentering US after sept 2021, you need to have fresh visa stamped on passport.In current situation , I am not sure if it can be done in Canada or Mexico. Ideal place is India.

Thank you. I’m renewing my passport now so I can go to India and get it done.