Neat idea. I just took a fast trip through Stax/Volt vol 1, 2,and 3, looking for songs that sounded "both mixed" rather than Neck or Bridge.
Here is my candidate, "Gee Whiz, It's Christmas," by Carla Thomas. It's on Vol 3 (of 9) so about 1963.
Now wouldn't a guy who lived in a recording studio know about jamming the switch?