Here are the results of totally non-definitive inquiry as to how common the term was prior to 1858. There is an on-line search engine that allows one to search scans of old newspapers. That is how I found in the 1868 article on the origin of the term. With that same engine, I searched U.S. newspapers for six variants of the term (jayhawk, jay-hawk, jayhawks, jay-hawks, jayhawking, jay-hawking) between 1776 and 1865. Prior to 1858, there were very limited usages:
o The Mountain Democrat in Placerville, CA had numerous hits for a voting precinct called "Jayhawk Flat"
o The 7/27/54 issue of the Ft. Wayne Times had a reference to "set of political jay-hawks who have stolen the public treasure"
o The 4/1/56 issue of the Ft. Wayne Daily Times has a column that appeared to be taunting a waffling politician, that included a phrase that someone had "jayhawked Fillmore and stuck John McLean under the fifth rib."

So except for a voting precinct at what I assume was a destination of the Jayhawkers of 49 and a couple of references by someone (or two) writing for a Ft. Wayne paper, there was hardly any published usage of the term (in the publications covered by the search engine).

The hits became much more numerous starting in 1858, with almost all of the hits in reference to the troubles in SE Kansas, and then expanding in various usages from there.

I believe this is supportive of a hypothesis that the term that was pretty rarely used until someone coined the term to describe the theft and plundering of the Kansas Troubles, with the capital J usage quickly attached to those affiliated with the Free State cause, and other uses (inlcuding the little J use in the Ozarks) branching off from there. They say where there is smoke there is fire, and there sure is a lot of smoke around Pat Devlin.

Kirby, I have a tremendous amount of respect for your insight, and realize that if you have an opinion to the contrary, the odds of my deductions on this being wrong are pretty high.