Here's what I'd do.
Call the front-line reservations number first..
Explain what's gone on.. See IF they'll give you the actual phone number to the airlines station office of the city where you disembarked..
Some airlines are loathe to give these local numbers out to the pubic as they prefer (for good reason) that all calls be centrally handled as local stations aren't staffed to handle the majority of inbound calls.
Once (and if) you get it, ask to speak to a supervisor who works either "gates", "customer service" or "fleet services".. all three of these titles generally means the employee has direct contact with the aircraft and it's servicing while on the ground..
Other titles like "cargo", "ramp" or "administration" tends to be people not directly involved with the aircraft itself or the passenger-side of the operation.
Normally most of these kinds of items are held at the airport where they're first discovered.. If they're not reclaimed after a certain number of days, then they're either sent to the airlines central baggage office for collection and mass disposition or are disposed of locally via an approved process..
As a passport is technically US Government property (they're issued to you, but remain DoS property) any US passports will be sent back to the DoS via their central office after a period of time.
I will tell you that IF it's found that most carriers will make at least some basic search to see if the name on the passport matches someone who has a current reservation.
Time is of the essence here as found items aren't usually held at the local stations for long periods of time due to physical space constrains (you'd be amazed at how many coats are left behind and the space they take up) and staff time involved in their collection and handling..
Best of luck.