I built a stand for mine as Hot Water suggested. I had never really thought about making my own... but of course it was extremely simple. 
I took a piece of Ash leftover from a window trim restoration project and drilled a hole to fit a piece of large gauge copper wire bent to form a hook. I've had the watch running and hanging in a vertical position for several days now and it seems to be keeping time just fine. I'm thinking that was my problem since I often just leave it laying on its back when not in use, though I keep it wound during the operating season.
Eventually I'll get around to giving it a proper coat of stain and varnish, but it's functional for now.
