Pretty sure you can do an in-place upgrade on the web interface of Hetzner
In place, you don't need to move data yourself to a new instance. You just choose the bigger plan in their web portal.
To answer my own question:
Scalability
Stay flexible with your Storage Share. Regardless of how your requirements change over time, you can upgrade or downgrade your Storage Share in a few quick steps and without worrying about data loss. Simply switch between the Storage Share package size you need by going to your account on the konsoleH, and then to "Account type".
Assuming you start with NX11, which has 1TB storage, and before hitting the limit you want to upgrade to the 5TB storage, NX21.
Can you just call Hetzner and tell them they should upgrade your NX11 plan to NX21 in-place, or will you have to order NX21 and then move all the data over to the new instance yourself?