G'day John, welcome to GPSPower. Unfortunately you cannot load a detail map image file using MapInstall as you've found. MapInstall will only load map data to it from .gmap folders on the PC.
The GPSMAP marine plotters and aviation units can have map info loaded in other ways. There are existing threads here, maybe a little hard to find easily exactly what you're looking for. This one is for units which can have gmapprom.img on the internal memory [that naming is for the preloaded map, the basemap is always 'gmapbmap.img' btw]: [Only registered and activated users can see links. ]. Maybe it will work if the map's renamed and the path etc in command prompt is changed to " <path to webupdater.exe> /i /autorun usb /upload_file rgn=10 path=<path to gmapsupp.img> " but i haven't tried it on a 276C. It works for old StreetPilots such as c3x0 with an SD and 26x0 which have a CF card.
I think your unit can only have the basemap on the internal memory, and the only detail map is on the data card named gmapsupp.img in a Garmin root folder. If you have a Garmin proprietary datacard writer, you can simply copy and paste the unlocked image file to it after renaming it gmapsupp.img. Otherwise, if you can connect to PC and use Updater.exe you can flash the map as an rgn file to region 10 [which in hex is 0A]. Use this tut as a guide by simply replacing the region as '0A' instead of '03' for the basemap: [Only registered and activated users can see links. ] The actual map img naming commenced with isn't important as noted in the thread.