I'm using RHEL 4.
The normal tail -F file_name*.log doesn't update when the time flips over to 0000.
Here's what I've tried so far from searching the web, both of which try to use nested tail:
tail -F $(ls -tr file_name*.log | tail -n 1)
tail -F $(fine . -maxdepth 1 -type f -printf "%T@ %p\n" | sort -n | tail n 1 | cut -d' ' -f 2-)
None of these flip over to new files. Plain old tail -F file_name*.log works in my small test cases (while the 2 above don't), but it doesn't work when I'm tailing a real log file.

tail -F filename.logis what you want. I am not following what you are trying to do with all of the extra nesting. – sparticvs Nov 3 '12 at 3:10