Using the PASSWORD() function

1. Add "passwordConverted" in the database, default into 0.
2. Select user from database based on username/email
2. Check if "passwordConverted" are 0 or 1

If 0:
1. Make a query with the password, if it's matches login user
2. Rehash password and replace password in DB.
3. Set "passwordConverted" into 1 in DB.

If 1:
1. Validate with password_verify, if it's matches login user.

According to Google, MySQL password() function are equal too:

PHP Code:

function mysql_password($string){
    $pass = strtoupper(
            sha1(
                    sha1($string, true)
            )
    );
    $pass = '*' . $pass;
    return $pass;
} 


If that's correct, you can do passwordConverted = 0:
1. If it's a match with mysql_password; login user
2. Rehash password and replace password in DB.
3. Set "passwordConverted" into 1 in DB.