You can think of a liner regulator (LDO) like the AMS1117 as a kind of intelligent resistor divider, which adjusts its value to maintain the output voltage stable. It is thus impossible to have more current at the output than at the input. In fact the output current will be slightly smaller by the value given in the LDO datasheets as "quiescent current", which is the current that the LDO needs for itself to operate.
If you need more than the 600mA at the 3.3V side you could switch to a buck converter. The operation of the switching regulators is very different from that of the LDOs and there you should think in terms of power. So, in your case, from the 5V*600mA=3W input power the buck regulator will deliver something like the 95% of it at best case, so 2.85W. This means that the available output current would be around 2.85W/3.3V=0.864A (best case).
I don't see at the moment any other possibility to have more current at your output. Of course you could always try to increase the input current.