If you do not know your George W's father's name, the best place to begin is to establish the county in South Carolina where your G.W. lived most of his life, and concentrate your efforts in that county - censuses, yes, but county library genealogical section, court records, books authored by South Carolina researchers in which excerpts are given from newspapers etcetera, and also engaging the services of a local researcher - all of which I did once I narrowed my search to a county. However, searching for G.W.'s father was quite another story that took several years and led all the way to Pennsylvania (as a result of, first of all a newspaper excerpt in one of Brent Holcomb's books which are conveniently indexed, and secondly a record discovered on my behalf by Mr. Holcomb in the 1838 patient records of a hospital in Columbia). So, it can be done, especially if you or someone on your behalf can research records at the South Carolina State Archives and/or the Caroliniana Library in Columbia.
Now, with regard to Laurens and Greenville counties. Both counties have very good libraries which house historical genealogical records. Concerning the census, my research shows a G. W. Cunningham, age 26, in the 1840 census of Greenville County, and he shows up again in the 1850 census in the same county at the age of 36. The 1850 census is the first census in which names of family members were given, as you probably know. His wife's name, however, is given as Martha, and they had six children at that time, none of whom was named Felix. In the same Greenville Cty census of 1850, there is a George, age 2, whose father's name is given as Henry and mother's name is Mary, along with a brother named Lawrence, age 1.
In the 1860 census of Greenville, there are two George's. One is age 65, born in Virginia. The other more nearly fits the time frame which you are researching. His name is given as George, age 5, which means of course that he would have been born in 1855. His father was Enoch Cunningham, born in South Carolina, son of William Cunningham who had been born in Virginia. This George was one of ten children living at that time. He is mentioned again in the 1870 census at the age of 16. (This was one of the first Cunningham families researched when I was looking for my G.W.'s father - and I thought I had the right one until . . .) To my knowledge, there were only two George Cunninghams born in South Carolina in the 1850s - my grandfather who was born in Richland County in 1854 and the George Cunningham born to Enoch in Greenville County in 1855. So, you might want to check the census of 1880 to see if the George Cunningham born in Greenville in 1855 is named in the 1880 census, either in Greenville or some other county in South Carolina. And, you might want to call the Greenville County Library or Historical Society to see if someone there will check their indexes for his name.Enough for now. Get back with me if I can furnish additional information. Good luck! And if you do not have access to the 1880 census, let me know and I will check on it for you the next time I go to the National Archives here in Atlanta.