I had thought that because the µCurrent can measure down to µA and nA level that it would be able to handle small mA values without any issue. I now understand that this is not the case, which makes sense to me and is fair enough.
Your problem is that it tries to cover 9 decades with only 3 current sense resistors (1mV/mA, 1mV/uA, 1mV/nA) so resolution at the bottom end of each range is poor. Your best option to get more resolution on low mA would be to source your own close tolerance shunt resistor and strap it across the input terminals.
You would need its value to be slightly high to offset the shunting effect of the internal sense resistor still being in circuit (set it to the lowest range so that the internal resistor is a high as possible - 10k I think).
With appropriate external resistors you can get 10mV/mA, 100mV/mA etc.
P.S. If you're not interested in measuring nA level currents then you could simply remove the 10k current sense resistor for the 1mV/nA range and dedicate that range to external sense resistors only. This would simplify external resistor values (in fact you could also use an external 10k resistor resistor to retain nA capability).