How does one update the wallet to reflect the latest Covid vaccination?

While the basic information that you are seeking is in the related article in the upper left corner, I want to expound a little bit on this and to share my personal experience just in case your experience is the same as mine.

If you have received all of your Covid vaccines from the same medical provider, it should be relatively easy. The QR that the provider sends you, or is posted in your online medical history, code should reflect the updated status of your vaccinations. If you haven’t received it, contact them. If you end up getting one of the shots outside of your medical providers system like I did with my third shot, you may have to contact your medical provider and ask for their help.

When I received my first booster (third shot) I scanned my CDC vaccination card and sent it to my doctor. I asked if there was a way for them to enter it into my records so that I could get an updated QR code. They obliged and I was able to add the updated information in the Health App and placed the card in my Apple Wallet. When I received my second booster (fourth shot) within my medical provider’s network, they updated the QR code and my records again and I added that information to the Health App and my Apple Wallet.

The link on top of this page which you had visited, before posting this question here, is shown under the "Related Article” that may have given you an answer.