I bring you routings which would not necessarily lower the price but possibly increase your chance for miles:

From JFK:

JFK-ORD-DFW-LAX-HNL-SYD-Adelaide
JFK-DCA-ORD-SFO-NRT-HKG-BKK-SYD-Adelaide
JFK-LAX-AKL-SYD-Adelaide
JFK-BOS-ORD-SFO-LAX-HNL-SYD-AKL-Adelaide

From DCA/IAD:
DCA/IAD-ATL-DFW-SEA-SFO-LAX-NRT-HKG-BKK-AKL-SYD-Adelaide

DCA/IAD-JFK-LHR-JNB-SYD-Adelaide

IAD-LHR-AMS-CDG-FRA-SIN-SYD-Adelaide

However, don't expect a small price, but miles and miles galore!