New phone transfers should offer a service that copies all your info to the new phone. If you can't do that, then try connecting your phone to a PC via USB. It should allow you to access the phone's data, and at that point it just becomes a game of finding the correct folder, making a copy, putting it on the PC, and then copying that over to your new phone in the same directory.
I'm pretty sure this game saves to the cloud, though, because I've done reinstalls and updates and it kept my progress.
Note: I play on PC, not my phone, but I've done finagling on my phone before and that's how I accessed the files.