My experience is that there is a particularly high shake up of availability from 10 days before departure right up to, and including, the day of departure. There was one occasion when I got an elusive CW seat to Sydney on the day of departure, and I know for a fact there was no availability on the night before. I think there is something in the terms and conditions that implies availability is only bookable 24 hours before departure, but in reality it can be done literally hours beforehand. On the bigger routes (JFK and the collective west coast cities) I routinely book slightly uncomfortable timings - with 2 single tickets - and swap the flights at the last minute as availability frees up.

Availability does open up over the weekends, but often on Mondays there can be big changes around 11 am UK time.