This worked great Michael, thanks for the good advice!

Hmm ... if I developed my own Intro component just a few days ago I would ask myself:
How did I make my Intro tab appear?
Ah yes, I used a callback:

ossn_register_callback('page', 'load:profile', 'com_intro_profile_intro_menu');

Okay, but do callbacks offer a priority parameter, too - similar to hooks?
(... opening ossn.lib.system.php again... inspecting line 278 ...)

function ossn_register_callback($event, $type, $callback, $priority = 200) {

Great, they do!
Then, all I need is just adding a priority value smaller than 200 to my callback to make it move left?

Yes indeed: Callbacks are using the same priority mechanism as hooks. Without explicit priority all callbacks default to 200, the sort order simply reflects the order the callback get registered (first in - first out). While explicit priorities smaller than 200 would move your callback more towards the beginning, and explicit priorities bigger than 200 more towards the end of the callback collection.