There are hundreds of country and western songs from 1922 and earlier. A few examples would be
Looking for lists of "folk," "roots," or "southern gospel" music will expand this list significantly.
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In the US, all songs published before 1923 are in the public domain (although sound recordings of them may not be). A good resource is linked below, but you can also look at the wikipedia pages for "1922 in music" and so on for a list. Folk Music, traditional songs, and hundreds of hymns are PD as well.