Can you assign a teacher to a new college when she/he's done with the rec?

Like what the title says…

say you assigned your teachers and they finished your rec already.
Can you decide to add another college after that and assign the same teachers?

I’m not sure how it works in your school.

But in mine, those letters are “requested” and not “assigned.”

The letters I’ve written can go to any and all colleges the kids apply to.

I’m pretty sure it all works the same way, since I’m using CommonApp.
Don’t teachers write one letter of rec on CommonApp for all the colleges the student is applying to?

If my teacher finished his/her LOR already, and I decide to add another college later on, will I be able to send the LOR my teacher had completed?

Yes, you can assigned teacher who has finished submission of LoR to a new school ou are going to apply. He/she does not need to do anything else.