An important step is to search the Tennessee Death Index online from the Shelby County Register of deeds. This will give you a death date to assist in your search. It can also indicate 'no record found'. The search then becomes more complicated, as you discover the place of death.
[All of TN can be searched for 1949 - 2009]

The death date for Corrine is about 2 days before the obit. appeared for a Corine C. (Mrs. Evins) Daniels, as listed in the Chattanooga Pub. Lib search results.

Tennessee death certificates before 1961 are public records, and available at the State Archives. After that, they are considered 'private', and more difficult to obtain. The obituary is definitely the way to go.

Another option is to see if a microfilm copy of the newspaper is available via InterLibrary Loan from one of the "Tennessee Newspaper Project" repository listings.

http://www.lib.utk.edu/spcoll/newspaper/tnphome.htm