If you're talking about being able to regain access with no local backups (even just a USB key sewn into your clothing) your going to need to think carefully about the implications if someone else gets hold of your phone, or hijacks your number. Anything you can do to recover from the scenario is a way an attacker can gain access. Attempting to secure this via SMS is going to ne woefully insecure.

That being said, there are a couple of approaches you could consider. One option is to put an encrypted backup on an sftp server or similar and remember the login and passwords, another would be to have a trusted party, say a family member or very close friend, hold the emergency codes for access to your authentication account or backup site.

Storing a backup somewhere is a reasonable approach if you are careful about how you secure it and consider if it meets your threat model. The backup doesn't need to contain all your credentials, just enough to regain access to your actual password vault, so it doesn't need to be updated often, unless that access changes. I would suggest either an export from your authentication app, a copy of the emergency codes, or a text file with the relevant details. Encrypt this with gpg symmetric encryption so you don't have to worry about a key file, and use a long, complex, but reconstructable passphrase. By this I mean a passphrase you remember how to derive, rather than trying to remember a high entropy string directly, so something like the second letter of each word of a phrase that means something to you, a series of digits that are relevant to you, maybe the digits from your first friend's address or something similarly pseudo random, then another phrase. The result is long enough to have enough entropy to be secure, and you'll remember how to generate it more readily than remembering the phrase itself. It needs to be strong as once an adversary has a copy of the file they jave as long as they want to decrypt it. Once encrypted, upload it to a reliable storage location that you can access with just a username and password. Now you need to memorize the storage location, username, password and decryption passphrase generator, but you can recover even to a new phone.

The second option is to generate the emergency, or backup, codes to your authentication account, or the storage you sync it to, and have someone you trust keep them, only to be revealed if you contact them and they're sure it's you. To be more secure, split each code into two halves and have each held by a different person.