Scott.
According to Burke's "Official Military History of Kansas Regiments" on page 24, Major Owen A. Bassett of Douglas County, KS was mustered 4 January 1862 into, I think, 9th Kansas Infantry. This unit was re-designated later as 2nd Kansas Cavalry. "The Official Army Register of the Volunteer Force of the United States", vol 7, page 345 states Bassett served in this regiment throughout the remainder of the war and was mustered out at expiration of term (3 years) on 6 January 1865.
I wonder if the other two cavalry companies were also folded into this regiment. Burke did not state that, but his narrative seems light on some aspects.
Offhand, I do not know the origins of the "Kansas Raiders" name, but I wonder if Woods' unit adopted that moniker while operating in Missouri, since many of his men were from Kansas.
I never read that any other unit used the term "Kansas Rangers."
Bruce Nichols