I want to thank Tom for his suggestion of backing up and then restoring. I was able to get the phone to go from 1.5-1.8GB of free space max to around 2.7. Sure, it's not a gigantic increase, but relative to the amount that was free before, it's pretty big.

I do like the backup and restore process that Apple has created. It works well, although I dislike all the steps you have to walk through when the phone comes back up. I'm not sure why it needs to set up the wallet and iCloud stuff again, when that should just be treated as settings and data that gets restored to the phone. If it's all encrypted, then why not just store all that too? And if it asks that because it wasn't set up before, then it should recognize that we didn't want it and leave it alone. Other than that, it's certainly easier to move your stuff from one iPhone to another than it is to migrate Android phones, and it's much more comprehensive.

We're going to look into an early upgrade for my wife, perhaps when the 7 gets released, because it really isn't sustainable to deal with this tiny amount of storage. I don't think we'd need more than 32GB, but maybe we'll splurge and get 64. 32 would already give us more than 16 times the space we've been working with up to now.