Yes, posting the vet clinic contact number is what I'd do -- they should be able to handle asking for proof of ownership.
The goal with a found dog is to cover the bases in the places an owner would look: the town animal shelter (if there is one), around the neighborhood, the newspaper, and online. Not all owners can be reached with any one method, so the more methods that you use, the better the chances.
We also just had a holiday weekend, so the chances of an owner being out of town and having a dog sitter are higher than normal. The vet clinic needs to give this some time before they adopt the dog out -- try to keep your focus on getting him home, not on rehoming, as a dog in this kind of well-cared-for shape (glossy coat!) probably has a decent owner out there missing their dog.
Last edited by Magwart; 04-17-2017 at 03:17 PM.