Do damages for an insurer's bad faith go beyond what I would be entitled to under the contract?

Yes. In addition to what the insurer owes you under the policy (plus interest), if the denial can be shown to have been "unreasonable," and you have been damaged by the insurer’s conduct, you might also recover "consequential damages" (monies you had to pay out-of-pocket because of the denial), and "extra-contractual damages" to compensate for mental and emotional distress, and, in some cases, "punitive" or "exemplary damages" designed to punish the insurer and deter it and its employees from wrongfully denying similar claims in the future. Not all jurisdictions allow all of these types of damages in all cases. It’s important to talk to a lawyer to determine what kinds of damages you can rrecover.