Saying that this was a freak series of events doesn't really excuse my failure to take even that remote possibility seriously and have it dealt with before leaving Safford this morning. Much of the route through Texas and New Mexico is extremely sparse in terms of services (particularly bike shops) and I'm really not pleased with myself right now.
But I suppose it was better that I learned this lesson here, where recovery was difficult but possible: I'm now back in Safford so I can have this dealt with tomorrow morning.
In other words, after riding ten miles out this morning before realizing I had to turn back, I walked the bike back those ten miles, then had to walk the bike the full length of this city four times before I found a motel vacancy. Walking that bike continuously for eight hours, much of that in desert conditions, and on shoes that are not designed for walking, was not an experience I'd call pleasant. As such I think it's pretty unlikely that the error in judgment which led to this is something I'm likely to repeat any time soon.