F3-Access doesn't deal with authentication, only authorization. So you can use whatever you wish in order to control access. It is very very flexible and a great plugin.
In your case, it's not clear what exactly SESSION.user is. Is that just a username? An array with the user's details? If it's an array with the user's details, then you could probably use something like SESSION.user.group
Personally, I generally use some kind of group type roles like "admin", "user", "publisher", etc. and then use that to control access.
Once you have defined which parameter you'll use to control access, it's just a matter of setting the appropriate allow and deny rules and use that to control access.
$access->allow('/admin', 'admin');
// (...more rules)
$access->authorize($f3->get('SESSION.user.group'));
You can also use a username or email in order to get more granular control if you wish. (although I wouldn't recommend this most of the times)
$access->allow('/admin', 'admini...@example.com');
// (...more rules)
$access->authorize($f3->get('SESSION.user.email'));
Finally, the plugin is flexible enough that you could also use a combination of both. Grant access on some areas according to the user's group role and on some other areas grant access to a specific username or email.
Just make sure that rules are defined for all cases and that you use the correct parameter when calling authorize().
The plugin's documentation on github is pretty complete and has everything I've said here but I'm sure @xfra35 will probably help you out some more when he sees this if you're still having issues.
Hope it helps. Cheers.