OK, in case anyone was following this one, I was unsuccessful in convincing the GPS (a Garmin 320c) to allow me to select from among more than one map file.
Both the existing US file, and the Mexico file I downloaded from the website Seppo mentioned (http://garmin.openstreetmap.nl/
) are named gmapsupp.img. I renamed them to gmapsuppUS and gmapsuppMX, respectively, placed them both inside the "garmin" directory on the SD card, and fired up the unit. A splash screen appeared and directed me to use the proprietary PC-related map uploader (MapWizard IIRC) before it could continue. I imagine this suggests that the unit's Flash ROM (or whatever) is programmed to look for the file gmapsupp.img, and nothing else, however similarly-named it may be.
This means that I'll have to carry the MX gmapsupp file on a separate SD card, and swap out the cards when I cross the border. Not the end of the world, for sure. And not that I'll be tracking it quite that closely, but I wonder if the unit will know to continue to record travel time and distance as if it were the same trip, even if the data card has been changed. We shall see! ETD December 4.