No. The severity of the issue depends on other details of your system, but in any case it is a bad idea.
If it is a salted hash, properly implemented (salt before hashing, unique random salt per user), then the hash alone will not expose the password. However, passwords are rarely changed, so the hash of the password would rarely change, so anyone who intercepted this hash would be able to use it to impersonate the user for an indefinitely long period of time.
If this hash is not salted, the problem gets a lot worse. A bad actor who intercepts such a hash would have a significant chance of being able to crack the actual password from it. It doesn't matter how technically irreversible the hashing algorithm is, there are huge databases of common passwords and their hashes and for a lot of users cracking an unsalted hash is a simple matter of a database lookup. If the user happens to use this password on many sites, as is common, this could turn into full blown identity theft.
The one part of your idea that is ok is using a token for authentication. That token should be purely random and strictly time-limited, however, with a new one generated each time a user logs in.