score:4
A number of travel insurers, eg. World Nomads (no affiliation), will quite happily sell you medical insurance per day and per country, so if you want peace of mind you could buy two separate single days of coverage from them.
That said, insurance makes the most sense when you can't cover the cost of catastrophe, but in Singapore the public medical system is both excellent and very affordable, even at the unsubsidized rates paid by non-residents. For example, if you're hospitalized for a week due to heart failure, even at the high end of the scale you're looking at a bill of only $3500 or so: https://www.moh.gov.sg/cost-financing/bill-estimator/total-hospital-bills/HospitalBillsDetail/F62A
...which compares pretty favorably with the 6 figure bills you'd be looking at in the US.
Upvote:3
If you're unprepared to buy for the duration of your trip (which may still be wise, if there are restrictions on the reciprocity between Australia and your homeland), the easiest option may be to get a credit card that provides travel medical coverage. This will generally cover trips up to a certain duration, so the length of your trip will matter. The cards usually include other benefits that are useful to travelers too, such as baggage coverage.
Even if you could buy the short-term coverage, there would be a minimum premium for each burst of time you need it, so you might pay as much for a single day as you would pay for a week or two of coverage; it may not be as cost-ineffective to insure the whole trip as you think.
Bear in mind, too, that if your aircraft stops in another country because of your medical emergency, it is possible you could end up in another country than these completely unplanned, as well, and need medical insurance there, too. This is also possible if your aircraft stops for mechanical issues or for any other reason.