I'm not sure how replacing the password itself with a hashed version would do you much better. Are you not still storing a vector of authentication material on the local host, that if compromised, resulted in access to the remote system?
I understand your concept of attempting to prevent attackers from brute-forcing the clear-text password with the original salted password on the host, but I don't see the good in that besides preventing an offline-attack to then be used online at a later date (possibly after the 1-month period that you described initially).
I am not sure of a solution and am newer to cryptography, however I thought it was worth pointing out that you basically just add an extra step to the process, that I believe provides you almost no added security.
This is my thought process:
If hash1 is stored on the computer, it might as well just be the salted original password, because while that could be recovered using a brute-force/dictionary attack, it is still the same idea where even if it is hashed again, it is still a secret that cannot be disclosed. This surrounds the idea that the salt is reused in both hashes, because I think it has to be in your use case.
I'd love to listen to feedback from others, as I am here to learn. Thank you!