Another idea would be to have a chatroom (eg Kraa) https://news.ycombinator.com/item?id=47872083 - and preload a bunch of people into it, and have each one notified when someone posts into it (that feature does not exist.) The 'leaf' could be called "Panic."
The issue is verification - a password seems the most obvious. Something you can remember once your phone is gone.
The other issue from reading comments below - you need to be able to walk up to someone and tell them, "I've lost my phone, do you think I can use your phone briefly." Alternatively, you could walk into a box-store (or library) and use one of their PCs.
Also, a way to get transport quickly would be important - perhaps that is the first thing you do when on another computer.