Thanks so much to everyone for their suggestions, especially to glex50 for the info on Virgin Blue. That's news to me, so it's great to know. I did look into RAR, TBU, and APW, but with the limited flight schedules (some routes are only once a week), none of those worked out. But I got a fantastic HNL agent who worked with me and put together the following:

Sep 16: ADL-AKL (spend Sep 17 in AKL)
Sep 18: AKL-HNL (arrives Sep 17)
Sep 18: HNL-ORD-JAX

So, while it would be great to take a visit to somewhere in the South Pacific, I was successfully able to put together an itinerary that gives me an official stopover in AKL and a 19-hour transfer in HNL, which works great...not sure why one agent said I couldn't even transfer there, but oh well. I'd love to stay longer in either AKL or HNL, but with all the talk of trouble getting *A awards, I'll take what I can get!