Why not book JFK - SIN and then call the next day and see if you can add the next flight on. As noted they have very low change fees and they might not even charge the change fee. I think that would be worth it.

Alternatively, you can book a stopover in SIN for $100 (I think SGD) on an award ticket.

Why not snag JFK - SIN and then do a stopover (you'll want it flying East to get to Singapore - we flew RDU - IAH - DXB - SIN earlier this year on EK F and were absolutely wiped out which has never happened on our Asia travel where we usually go West to get to Asia). You can have a night in Singapore (which is worth it) and then should have no problem getting a flight down to SYD (and you can choose from any of the options the next day or two days later).

I have 2018 targeted to do the exact same thing and that's $100 stopover is our plan regardless of whether availability is there to do it all on one go (just to break it up and also because you can't fly through Singapore and not stop and go get some food at the Hawker Centres