How to move “Expired posts” to the end of the loop?

When a post is published into a category, lets say https://www.adoptiicaini.ro/cat/bucuresti/ the post appears first in the loop, but after the post is modified and added to another specific category (named Expired) I want the post to be moved to the end of the loop.

It’s like sorting posts by category, but all I want is to move to the and all posts that has the category Expired checked. This have to work every time, for all categories alphabetically after or before Expired.

I don’t want to exclude them like this:
https://codex.www.ads-software.com/The_Loop#Exclude_Posts_From_Some_Category
because the categories might end up empty at some point. Instead I want them to remain in the category loop so that, in this way, people from Google don’t get 404 errors saying page 3 of the category does not exist anymore.

I need to do this for a website with dogs for adoption ( https://www.adoptiicaini.ro/ ) , so anybody who loves dogs please give this a look ??

Thank You!