Can I re-enter Thailand with the same 60-day tourist visa after leaving for Australia?

Unless you have a Multiple Entry Tourist Visa that is valid for 6b months, no, you cannot use that visa anymore. You need to apply for another Single Entry Tourist Visa if you want to enter Thailand for 60 days again.

Check your visa sticker if it says single or multiple entry, or stamped as "used".

you'd have to LOOK on that visa and see if it was a single or multiple entry one

A single entry tourist visa is valid for 3 months from the date it is issued and you can use it ONE time to get in on a 60 day stamp

A multi-entry tourist visa is valid for 6 months from the date it is issued and you can use it as many times as you want to enter thailand during the 6 months it's valid for

SO to answer your question IF you had a single entry tourist visa, entered and got the 60 days you're done.

You can get the 30 day extension at the immigration office

but

then you'd have to exit the country and either come back in without any visa (on a free 30 day entry stamp)

OR

go to a thai consulate and BUY a new tourist visa