The only other George W. Hutcherson I can find a record of was a member of Co. C, 16th Arkansas Infantry, and was killed in action at Corinth on October 4, 1862. This company was recruited primarily in Johnson County, Arkansas.
Since your George W. Hutchison lived in what is now Baxter County, you might want to check some Missouri regiments. Some of the Missouri regiments recruited in northern Arkansas, and in fact some of the component companies were comprised primarily of Arkansans.
I checked the 34th Arkansas casualty lists for Prairie Grove, Helena and Jenkins' Ferry (the major engagements in which the 34th Arkansas participated), but Adrian Hays is not listed. He was dropped from the rolls before the February 28, 1863, muster roll, with no indication as to why.
Regarding the 14th Arkansas Infantry, the regiment was organized in July 1861, but the earliest surviving muster rolls cover the period May 1 to June 30, 1862 -- practically a whole year's worth of muster rolls missing. As a result, some of the original members of the regiment are lost to history. All you can do in that case is to fall back on secondary sources. Contemporaneous letters and diaries are preferable, and you seem to have such letters.