My husband's ancestor (different surname) married a woman that had had a son born to her, she was not married to the father. After they married, they sued for support, a "bastardly bond", in a TN county court. The man was made to pay a bond to the court. You might check the Arkansas County courts for any such suit and bond for Charles Jesse. According to WWI Draft regisration card it looks like he stated he was born in Kerner, Arkansas, his WWII Draft Registration Card has born Boone County, Arkansas.
Boone County was formed from part of Carroll County in 1869. If his mother was born in 1867 she was born in Carroll County. There is an Alford family from TN with children born in Arkansas in Carroll County. A Franklin L. Alford filed a deed in Harrison, Boone Co land office dated Feb. 1888 for 40 acres located in Carroll County. 1900 census this Alford family is living in Osage, Carroll County. This Alford Family is included in the Group One Alford results too. On the Alford DNA page, group one, click on "John Alford born about 1786 in North Carolina" Franklin L Alford is on page 5 of the family tree. My husband's Alfords of this family tree went to MO from TN. His nearest ancestor is an Alford female.
Have you contacted the Alford American Family Association?