Really...good 'ol stalking the website is my go-to method. Once that 30 day mark hits, people will start fine tuning plans, so I've found that last month to be the best time to stalk.
I've had things open up at all different times of day, but tend to have better luck between 6-9 am on weekends. It's like some people have some great dining epiphany on Saturday night and put it into action on Sunday morning. Not saying it always happens like that, but it's happened to me a lot.
Beyond that, the day before has kind of become the magic day to pick up all kinds of things from people dropping stuff to avoid no-show fees. I've seen everything from well timed 'Ohana and Chef Mickey's to fireworks timed California Grill and Rose & Crown with this day before approach. Next time we visit, I plan on doing this for the majority of my dining since we're past the days of we MUST do this.