I think when you reconstitute your private keys on a new cold wallet, your old wallet becomes obsolete and can be safely discarded. Your funds will now be stored on your new wallet, which is secured by your new private keys and passphrase.

It is very important to note that you should only discard your old wallet once you have you confirmed that ll of your funds have been successfully transferred to your new wallet. You should also make sure to securely erase any sensitive information that may be stored on your old wallet to prevent unauthorized access.

Make sure to store your passphrase in a secure location, such as a password manager or a physical safe, to prevent it from falling into the wrong hands.