If the old data cannot be accessed by either the remote handlers or the new add-in, you can create an update of the old add-in that adds a data export capability and a UI for the capability.
Users would first update the old add-in, and then use it to export the data to a location where the new add-in can access it.
However, data that would still be useful to users after the add-in is uninstalled should not be deleted.
Websites created by an add-in, other than the add-in web, should usually be considered data.
To make a change of type, you need to migrate from an old add-in to a new one.