If you drive through the night, your low-cost sleeping options are significantly expanded. In the afternoon, you can doze for several hours in just about any small hospital's main lobby, for example, or a small-town public library's reading room.

I haven't actually done this, but I'm pretty sure it would work, especially if you wear something that says "comfortable but not seedy", e.g., khakis and a shirt that buttons down the front and has a collar.