"Going Loud" does not result in the "Blown" condition. The Blown condition is (DtD p.119):
The God-Machine knows your Cover is just that. It might not be actively hunting you, not yet anyway, but you can’t hide from it any longer. Even if you have multiple Cover identities, the God-Machine has your frequency and the only way to lose it is to destroy the compromised cover.
In this case your Cover still exists, but the GM knows that it is a Cover. In contrast, "Going Loud" consumes your current cover to fuel a variety of effects (DtD p.196):
Going loud allows the demon to completely heal any lethal or bashing damage and shrug off any Tilts affecting her. In addition to this, when the demon goes loud, her current Cover is lost; she replenishes all her Aether, her Primum is set to 10, and she gains access to every Embed her Incarnation gives her affinity for along with all Exploits.
The DtD manual refers to burning the demon's Cover completely to fuel all of these effects. It's not a question of being able to track a demon back to their original Cover, since that Cover no longer exists. Without an additional Cover, the demon is stuck in its demon-form.
Using the "Terrible Avatar" Exploit may split the demon from its Cover, but subsequently "Going Loud" consumes the Cover to fuel further transformations.