Hoyt,

According to the definitive book "Tennesseans in the Civil War" published in the 1960s no unit exists with the name 3rd West Tennessee Cavalry. There was a 1st West Tennessee (later renamed as the 6th Tennessee Cavalry US) and a 2nd West Tennessee Cavalry (later renamed the 7th Tennessee Cavalry US). I find nothing for the 3rd in this book at all.

It seems they never formed despite the capture noted of some recruiting officers.

Greg Biggs