Does it typically become easier to get walk-up permits as it gets later in the season?
Yes. Once school is back in session, it shifts. Try to avoid Fridays & weekends. Mid-week during September is an excellent time - the weather tends to be stable, and the hikers have a higher level of experience than in summer. The days are shorter though, so if you're dayhiking and want to get out in daylight you will need to start a bit earlier.