Higher precision with less digits?
No, you never attempt to extrapolate beyond the displayed digits of a single reading.

In fact, the situation is much worse than you think. A "realistic" (as you say) ADC has several sources of error, which are usually specified in terms of offset error, gain error, and non-linearity. You must read the datasheet for the ADC or meter that you use. If you do, you will probably discover that the accuracy of the meter is something like \$\pm 0.005\$ mA for an indicated value of \$0.002\$ mA. Yes, the error is typically much larger than the resolution. Whenever you record or report a measured value you must also report the error in that measurement.

I should mention that if you understand the statistics of the noise in the readings you may be able to oversample to achieve higher effective precision. A good engineer doesn't do this by just "estimating" by eyeball, but by actually recording a number of measurements and averaging.