When a session is closed, for example, when a timeout triggers or when the user clicks the logout button, the session should be terminated at server side so the session cookie is invalid.
I think that it could be useful to also set to a blank or invalid value the session cookie in the browser, even if it is a temporary cookie. For example, when you click the Logout button the server could answer with a Set-Cookie that sets an invalid value for the cookie.
Does this measure improve the security of the site or is this "too much" having into account that the cookie is temporary and the session is closed correctly in server-side?