The Chattanooga Public Library may be open during the hours you need. All 4 locations have free wi-fi and computers with Microsoft Office available for public use, but if they are busy, you will be limited to 2 hours on the computer. A library card or a single-use guest pass ($1) is required to use the public computers. To see branch locations and open hours, go to http://chattlibrary.org/locations.
You can also try the UTC Library as they're sometimes open when Chattanooga State isn't. UTC's library has guest "Quick Look Up" workstations that allow access to the library's website, databases catalog, online journals. No login is required, but note that there is NO Microsoft Office software (Word, Excel, etc.) on the guest workstations. Information on the guest workstations is available on UTC's Alumni and Visitors page: http://www.utc.edu/library/services/alumni-visitors/