Why does the New email signature only shows when I click the Xink 365 add-in?

By default, the email signature will show when you click New email.

If you only see an email signature when you click the Xink add-in, check if WebView2 is installed as per Microsoft prerequisite for add-ins:

https://learn.microsoft.com/en-us/office/dev/add-ins/concepts/browsers-used-by-office-web-add-ins