To be specific about my claim: throwing away the earlier version and replacing it with the argon-wrapped version cannot make the user's original password easier

To be specific about my claim: throwing away the earlier version and replacing it with the argon-wrapped version cannot make the user's original password easier to recover, because it adds no new information about the password. To convince yourself of that, imagine that there was some function `argon(hash)` that made the password easier to recover. Why wouldn't an attacker run that function themselves? In the very worst case a bad hash function that takes one second to run could only speed up the attacker's job by one second. Wrapped hashes could have other side effects for probing non-hacked sites -- for example, attackers could probably figure out which chain a user has based on timing, which would let them narrow down a bit when the user last logged in. Hash chains could also theoretically reduce the total output space for the final hash, making it easier to brute force a password through the login form of a non-hacked site. I don't consider either of those likely to matter, but you might.