Hooking up the batteries in parallel won't increase the current; it'll just last longer, and less current will run through each battery.
There are 2 ways to increase current in a circuit; either increase the voltage, or decrease the resistance. Unless you have a resistor in series with whatever you're trying to power (eg. LED) that you can swap for a lower value, you'll have to increase the voltage by hooking up the batteries in series.
Be careful not to increase the voltage beyond any of the circuit devices' ratings. Devices may also have a max current and/or max power that you do not want to exceed.