The census taker handwriting is completely non-consistent. You will notice even when he spelled "Little" on the next two lines that the "L" is completely different. You can also notice that the "o's" and "a's" in the middle of the names throughout the pages are completely different with some a's trailing at the bottom and some trailing in the top. On the 2nd page after the page her name is on you can see the surname of "Linn" with a completely different "L" written. In fact, her Lackey name in the 1860 census is referenced as "Sackey" on Ancestry.com.
Since the writer is not consistent with his writing I suggest that the intended name spelled here MAY(?) be "Lackey". There are many cases where I have searched my Lackey surname and found the "L" looking like an "S". And seeing his "a's" and "o's" and what looks to be an "o" in that name could actually be an "a" is probative.
If it does not spell "Lackey" then I would agree with others that it is more likely to be "Sockey" instead of "Socially".
Someone with more handwriting comparison skills may be able to adequately answer this.
Coincidentally, I have several families in the Randolph Co. ILL area, including Sparta. My names there are Gray, Pillers, Bilderback, Steele, Vickers and others.
I have an old book my gg-grandfather bought from RG Shannon in Sparta, ILL in 1833. (This is penciled in the cover pages of the book by him). RG Shannon owned and operated the first trading post in the area after George Rogers Clark and his men took possession. My gggg-grandfather, James Francis Pillers was one of Clark's 200 original men.
I believe one of my family members also married into the Edgar family. Would have to search my data base to find which one.