Thanks for all the suggestions.
I've cycled in Cuba and Costa Rica in the past. Either of those or other places in Latin America would be great — if I score a deal on airfare. (If I had not spent a couple of weeks in the Yukon and Northwest Territories this summer, I wouldn't be looking to the same degree at a low-budget trip.)
Another option could be a loop from Las Vegas to Los Angeles and San Diego. Flying in and out of Los Angeles or Las Vegas is quite reasonably priced.