New and old passwords on same MySQL database?
It is possible before version 5.7, where it has been disabled.

To set up an old password for a user, use command:

SET PASSWORD FOR 'some_user'@'some_host' = OLD_PASSWORD('mypass');


More information: Password Hashing in MySQL.

You can enable/disable old passwords globally or per session. On a server where they are enabled globally (i.e. the server was started with old_passwords = 1), run this:

SET @@session.old_passwords = 0;
SET PASSWORD FOR 'test'@'localhost' = PASSWORD('mypass');


to disable them for a session. The PASSWORD in the above snippet will return a password in the new format, affecting only the current session.