I'm no expert, but I did some reward travel to Oz (IAH-MEL; CNS-IAD) earlier this year and even six months out when I booked it, and even though it was the low season, availability wasn't great.

A cursory search of December and January suggests that even with different origins and destinations (e.g. reward ticket terminates in AKL, SYD, CNS, BNE, etc. or ticket originates at LAX or SFO, under the assumption you could cover the domestic ticket on either end) didn't help availability much. A smattering of days with standard rewards would open up, but I didn't find any saver rewards.

Sorry I can't offer more help. Best of luck!