Getting MyBB to eat a Yummier Cookie?

[Firefox Wins]

Edit #2: If there is a fairly easy answer to "the green section" (below), then I'd be glad to learn; If not, "the other way" is at the bottom of this post.
^^^^^^^

Thanks for the quick answer, Zinga:

I lost you...
(For example), I'm logged into MyBB and close the browser, then tomorrow I return and I'm still logged in.
The moment a session is re-established and MyBB "decides" I'm still logged in (or just logged in, whatever) --> How to get the system to accept a different cookie?*

*Edit: That was poorly worded:  "How to get the system to accept a cookie w/ a different name, not myBB's standard cookie name?"

Are you trying to do SSO?
Yes

...Or you can just simply send the MyBB authentication cookie once the user has authenticated.
No, when "Secure (outside) cookie" expires then the end-user needs to be automatically logged out of MyBB, that is why I don't want to use the MyBB authentication cookies.

The other way:
If (user 'is logged in' && 'currently active*' && $sso_security_check_time = 1*)
{check sso_cookie ;
.....
}

* If someone is still logged into $global_sso, then their session (and MyBB cookie) is not affected;
If someone chooses to be logged out from $global_sso, then their MyBB cookie will also be logged out / expired.

Radically changing the system (like I assumed would need to be done) is too hard, especially when there is a much easier way.  
Thanks again.