I've had the opposite problem.

I was traveling with a large group of people (8 or 9 of us, I think, headed on a wine tasting trip in California), and in the rush to detrain at a brief, unstaffed station stop and offload everyone's luggage, one of us grabbed a 6-pack carrying case of wine from the downstairs luggage rack. We didn't realize it wasn't ours until we popped it open as we were loading it in the rental van a few minutes later and noticed that it was filled with a bunch of cheap wine (definitely not ours!) and a half-drunk bottle of Skyy vodka. Of course, by that time, the train had already pulled away.

I immediately called Amtrak and they supposedly documented the incident and "sent messages to all of the stations up the line" so that the rightful owner could claim it and Amtrak could get it to them. Of course, being that I was at an unstaffed station, I couldn't simply drop it off with Amtrak personnel (and the next stop was more than an hour up the line, so I couldn't chase the train), but I offered to drive it to the nearest staffed station (~30 minutes away) if the lost item was claimed so it could be carried to its rightful owner.

Despite giving my phone number, I never heard a thing back. After waiting a month, I gave it away.

It seems Amtrak doesn't have very good procedures or infrastructure to handle lost and found claims.

For one, they need to have a better, more centralized way of filing and processing claims. "Sending messages" to stations up the line seems like it leaves a lot of potential cracks in the process.

Two, they need to have a way to contact on-board crews to handle claims filed before the train reaches its destination. In my case, if Amtrak had the ability to contact the on-board crew, the rightful owner could have been located and notified and I would have dropped it off that same evening, and it would have been reunited with the owner the next day. In the OP's case, the car attendant could have looked in the closet, found the jacket, and ensured it was secured and sent to the right department (baggage/shipping) so it could make its way back to the OP.

I understand that using BNSF/UP/CSX/NS/etc. radio time to handle stuff like this may not be the best way to deal with it and would be logistically difficult. However, conductors all now have iPhones; perhaps this messaging functionality should be integrated right into the app that the conductors use to scan tickets and monitor train status and things. But at the very least, those iPhones have phone numbers (and conductors do call into dispatch regularly, especially in dark, non-Amtrak-owned territory to report station departures and delays and things), so they are reachable by voice.