In addition, I would go through the bibliographies of all the books on Jennison, and Jennison's Jayhawkers. Maybe the memoir show up in one of them.
Also, I wouldn't overlook name variations--Jack; Chiles, Child.... The NPS site is down as I am writing this, but wouldn't hurt to give the name variations a run through once it is back up again.
Since when you consulted it earlier it indicated he came out of the 15th Kansas Cavalry as a sergeant, it appears he may have served in a subseqent unit--perhaps one of the post-war cavalry units raised to deal with Indians (18th Kansas Cav., 19th Kansas Cav.). They were formed two and three years after the war, but followed the wartime numerical scheme. I have a major project going on right now with the 18th, and will look through my materials to see if anything pops up.
I can't identify Childs in a post-war census, but maybe his National Archives military and pension files will shed some light on a post-war residence. With it you might be able to locate materials in county historical society archives.