The idea that it is coming from approximately the center of the engine compartment and gets quieter when the engine is warm sounds like it could be related to an engine vacuum leak.
Do me a favor. Do a quick inspection of the engine vacuum hoses along with the air intake tube, and intake manifold. Make sure nothing is disconnected, damaged, or dry-rotted.
Also, if it is possible for you to record it and upload it for me to hear, it will be helpful. In addition, here is a link that explains how to locate and repair an engine vacuum leak:
Let me know.
Tuesday, January 24th, 2023 AT 2:55 PM