Although my family never stayed at one of the motels in question, the image was a memorable and singularly inviting one.
Search terms, on both Google and Google Images:
bear + hammock + hotel
bear + hammock + motel
bear + hammock + hotel + billboard
teddy + bear + hammock + motel + billboard (This produced the Teddy Bear Motel in Whittier, North Carolina--not the one I had in mind.)
teddy + bear + motel + hammock + motel + signs
sleepy + bear + motel (This produced references to the Sleepy Bear Motel in Gatlinburg, Tennessee--also not the bear I'm looking for.)
sleepy + bear + hotel (This combination yielded links about Travelodge and its mascot Sleepy Bear. This is the warmest I've gotten, but still not the bear mascot I had in mind, who was rendered in a more minimal cartoon style and wore a straw hat rather than a nightcap, coming across as an adult rather than a cub.
tasty + world (Tasty World was a restaurant chain, attached to a hotel chain, covering the same region and period; I had forgotten which hotel chain it was associated with, and performed the search in hope of a possible overlap. The late Athens, Georgia nightclub is unrelated.)
tasty + world + hotel (This resulted in listings of hotels in the vicinity of the above nightclub.)
"tasty world" + restaurant--established that the old Tasty World restaurant chain was affiliated with Days Inn; there seems to be a surviving specimen in Hahira, Georgia as of May 27, 2011.
bear + hammock + billboard + retro
bear + hammock + billboard + retro + motel
"teddy bear" + hammock + billboard + retro + motel