H1B-CapExempt to H1B-CapSubject, do I need new stamp?

I’ve been working on a H1B-CapExempt visa(with a non-profit employer) with a valid stamp Aug 2019-Aug 2024. Recently, my H1B-CapSubject visa(with a for-profit employer) got approved (got success in lottery and now have an I-797 approval notice from Oct 2022-Oct 2025).

Do I need to get a new stamp on my passport? or use the valid old stamp?

Most post I have read are on transfer from one H1B-CapSubject employer to another H1B-CapSubject employer. In this case, one can use the old valid stamp. My question is different since I'm moving from H1B-CapExempt employer to a H1B-CapSubject employer(I didn't find an answer to this case on internet! ).

Visa is only a travel document. If your I-797 approval notice is with I-94 ( starting 1st October) you dont need to go for visa stamping unless you want to travel outside the US and your visa is expired. If no I-94 attached then you need to go for stamping to activate your cap-subject H1B.
Consult your employer’s immigration lawyer.