The sound-deadening panel that is sometimes attached to the inner hood panel is the:

The sound-deadening panel that is sometimes attached to the inner hood panel is known as the insulator pad. This component plays a crucial role in reducing engine noise and vibrations from the engine compartment, providing a quieter cabin experience for the vehicle occupants. By absorbing sound and vibrations, the insulator pad helps enhance the overall comfort inside the vehicle.

The headliner, while also involved in noise reduction, is primarily the material that covers the interior roof of the vehicle, and it does not serve the same function as the insulator pad attached to the hood. The muffler is a component of the exhaust system that reduces noise produced by the engine's exhaust gases, whereas the vibration damper is generally associated with components that mitigate vibrations in structures or mechanical systems, rather than specifically addressing sound within the engine compartment. Thus, the insulator pad is the most accurate term for the sound-deadening panel related to the inner hood area.