There are alot of Thomas Cook's in Georgia around that time (1866).
This could be a possibility: https://familysearch.org/pal:/MM9.1.1/FWS1-XQT : accessed 19 Nov 2012 however it doesn't include an image or a region.
Here is another:
"Ohio, Deaths, 1908-1953," index and images, FamilySearch (https://familysearch.org/pal:/MM9.1.1/XZHN-KS7 : accessed 19 Nov 2012), Thomas M. Cook in entry for Marie Skidmore, 1937; citing reference fn 59450, Ohio Historical Society, Columbus, Ohio. Rome, Georgia is in Floyd Co. but it may be close to Fayette Co.
You could also try Fayette County Historical Society or geneology group they may have access to regional history that is not online.