No, fiat money is what we have NOW.
You are still mistakenly under the impression that the paper you hold in your hands is money. It's not, its cash. And all it does is represent fiat money. The actual money is nothing but numbers on a computer.
Our money became fiat money the instant we went off the gold standard. Whether we use paper cash or not, our MONEY is fiat.
Here, this should help clear up your misconceptions about what fiat money is.https://en.m.wikipedia.org/wiki/Fiat_money