NB: I've never played with configuring it.
I can't find anything on configuring it though, that only mentions for first-time contributors as you say. I have a 'contributor' badge for pretty minor contributions to all sorts of projects, doesn't necessarily mean they should trust me any more than a first-time contributor! (I can be trusted! I just don't think my contribution to some of them is too much of a barrier for a malicious actor...)
It's also configurable at the organisation and enterprise level, if those are relvent to anyone.
Organization settings: https://docs.github.com/en/organizations/managing-organizati... Enterprise settings: https://docs.github.com/en/github/setting-up-and-managing-yo...