A second chance is a gift. What you do with it is the thank-you note.

Gratitude that lives only in words is thin. Being given another attempt puts a small debt in your hands, and the only currency it accepts is behaviour over the following months. Saying sorry again is easier and does less.

If someone let you back in, resist the urge to keep apologising. Pick the specific thing that cost them something and make it visibly different, without narrating the effort. They will notice, and the noticing will do more for both of you than any conversation about it would.