Apparently the Superior Match Company produced a whole “Americana” series of matchbooks (also known as “Scenic America”) with similar scenes. You can read about it here.
Although only a few letters of the artist’s name were legible on the painting, by trial and error I managed to come up his full name – J.F. Kernan – and eventually the name of the hunting painting: It’s A Bargain.
Here’s a nice print of the whole scene.
It must have been a popular painting, with prints of it all over the internet, as well as a puzzle.
UPDATE (September 2, 2015)
I noticed another Grandview Motel matchbook cover (below) from the J.F. Kernan series on Ebay today. This painting has its actual name – "When Seconds Count" – printed on the matchbook cover.
It too received the puzzle treatment.