Can you recommend a good savings account?

OP, I realise you said you want to stay away from the big 4 but NAB may be worth looking at. They have an everyday banking account that is fee free where you can get an eftpos or visa debit attached. they also have something called an isaver that is an internet only based account but attracts a higher interest rate. And you don't need a home loan to get the free account.
 
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