Run upstart after all other init scripts have started
I ended up figuring this out. CentOS6 uses upstart as its init program, but one of the scripts it initializes in /etc/init is rc.conf, which starts up the old-school rc scripts. So if you need your program to start AFTER those you can put:
start on started rc
stop on stopped rc
in your upstart script and you should be good to go.