I'd tried the OR Atlas, no joy.

There may be some hidden hints in this Goggle search:
https://www.google.com/search?client=safari&rls=en&q=battle+of+salem+arkansas&oq=Battle+of+Salem%2C+Arkansas&gs_l=serp.1.0.0i22i30.4140.11852.0.13735.25.22.0.3.3.0.111.1935.20j2.22.0.msedr...0...1c.1.64.serp..0.25.1939.c0inxb4_3G8

Here’s one:

Battle between the Confederate and the Union soldiers in an area along the South Fork River about a mile northeast of the Salem square.
http://www.waymarking.com/waymarks/WMKMHK_Battle_of_Salem_Salem_Ar