Does FedEx use other airlines?
There are regular flights between Singapore Changi and Sydney operated by FedEx Express. Here is a list of past flights from Singapore to Sydney (FDX9075):

(flightaware.com)
There are regular flights on Monday, Tuesday, Wednesday and Saturday. You can see that there was no flight on Monday, 05 October. This is likely because the aircraft coming from Sydney (FDX9077) diverted back to Sydney on Monday:

(flightaware.com)
I don't know the reason for the diversion, but as you can see, there is an MD11 enroute to Singapore right now. It is therefore likely that the same aircraft will depart Singapore back to Sydney later today as FDX9075, possibly carrying your parcel.
Update: Return flight is in the air right now:

(flightaware.com)

To answer your question about using other airlines:
Normally, FedEx does not use other airlines for international flights. They operate their own network of flights with their own airline FedEx Express. In these special times, it is quite possible that things have changed1, but I could not find anything official.
FedEx is however using a lot of other airlines for local transportation. They are called FedEx Feeder airlines. Wikipedia has list of them. They only operate regional aircraft (most of them ATRs).
1 Normally, a lot of air cargo goes on passenger aircraft (below the passenger deck). Due to the COVID pandemic, passenger travel is significantly reduced right now, which means more cargo needs to be transported on dedicated cargo aircraft than before.