Well, more elaborate chords are typically used by groups other than Bill Monroe's, particularly those backing up Texas-style fiddlers, who do great things with this tune. (Listen to Sam Bush on "Poor Richard's Almanac", for example). They use a nice descending chord run the B-part, which is distinctive (this can also be played as a bass run). Also, they tend to play this tune with three parts, not two. Finally, that nice major/minor tension on the A chord in the A-part is often dealt with by playing a backup chord that's missing the 3rd altogether (called an "A power chord" by some rockers; or an "A drone chord" by some bluegrassers). Or, you can swap an A minor for the A major.
Try out these chords, for example, which have the chord run in them (each chord is two beats long, the slash means to repeat the previous chord for two beats)
A-Part (x2)
A / A G A / E7 A (Note: use an A-chord that's missing the third, like 2200 on mando)
A / A G A D E7 A
B-Part (x2)
A / A / G / E7 / A G#m F#m A D A E7 A (Use a normal A major chord)
C-Part (x2)
A / G / A / E7 / A / G / A / E7 A
Watch Tony Rice's hands in this video to see the chords he's playing! Nice arrangement!!
And for the full, Texas-style longbow treatment (slower pace, swingier chords), listen to Texas Shorty playing here, and how greatly the backup differs!
In case this is helpful, here's a mandolin arrangement of Dusty Miller, inspired by Sam Bush's fiddle playing. The TablEdit file includes a guitar and bass backup track, so you can just turn the mandolin lead off and use that as a backup track to practice, if you like. I am assuming you have access to TablEdit or the free TefView player.
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