What you should do, is not just store phone numbers, but a list of name and number for each contact. Basically, you will get a list of lists stored in your TinyDB.
To set your listpicker, you would have to present each entry into a readable form, how to do that is shown here:
So, you do not have to start over, but at every step you have to do something to work with a list (of name, number) instead of just an element (number).
Cheers, Ghica.