In app photo transfer without taking them out?

I have private photos in an app that i dont want in my library so i store them in the app. I'm thinking about getting a new phone so is there a way to get them to my new phone safely without having to take them out and backing them up to my library or something. Or can my phone company just take out my sim card and put it in my new phone and it will all be like my last phone all as is?