I was finally successful in finding Hiram Barnett in the 1860 U.S. census. In a line by line search in Hardeman Co., Tennessee, I found the H.J. Barrett household that matches the family of Hiram Barnett. Here’s a summary of what appears on the census:
Barrett, H.J., age 51, M, farmer, personal estate $200, born Ky
Barrett, Sarah H, age 44, F, born Tenn
Barrett, Jones M., age 22, M, farm labor, born Miss
Barrett, Geo W, age 17, M, farm labor, born Tenn
Barrett, Sarah F, age 15, F, born Tenn
Barrett, Elizabeth, age 12, F, born Tenn
Barrett, John N., age 9, M, born Tenn
Barrett, Benjamin M.,age 4, M, born Tenn
Barrett, Evaline, age 1, F, born Tenn
In addition a Public Member Tree (“My Forebears Since Jamestown by meggars55) at Ancestry pointed me in the direction of additional records:
Sarah H. Barnett and some of her children appear in the 1870 U.S. census in Tipton Co., Tennessee and in 1880 they appear in Greene Co., Arkansas.
It appears that Hiram Barnett may have died sometime between 1860 and 1870, probably in West Tennessee and his family went on to settle across the Mississippi River in Greene Co., Arkansas.