Unfortunately yes you then need to continue to Sydney or Japan (believe these are the only oneworld flights out of HNL) -> I would however observe that this route (which I'll take for the second time in a month on a DONEx) is more pleasant as you have a long haul plane and service, including access to the Flagship lounge. And if collecting segment based earnings (alas for me, now gone due to the BA changes) it is 2x what you'd normally get on, for example, DFW to NRT.

So your ideal route would be something like this for your six segments (think this would work?), where XXX is as far away as avoiding the trans-continent restrictions:
JFK-LAX (transcontinental), LAX-XXX-DFW-ANC-DFW, DFW - HNL

Even better if you can route the opposite way, as JFK-XXX-DFW would give you a lot longer flight options than from California per this table:

Oneworld Explorer restrictions across the continent