Lowering current measurement range?
Use a shunt resistor, but use a large enough shunt resistor that you can see a meaningful voltage at the currents of interest.
As in your original experiment, this shunt resistor may present too much of a burden voltage to get through some more power-hungry tasks you need to pass through to get to the phase of operation you are interested in measuring, so temporarily bridge the resistor with a clip lead until you get to the situation of interest.
In some cases it's also useful to add a capacitor to the circuit, possibly by putting it across the resistor.
These techniques can be extended to some types of time-varying behavior by using a scope instead of a simple voltmeter.  Sometimes it's useful to use a fairly high resistance shunt to determine the actual sleep current, then a lower one and a scope (or MCU datalogger) to determine the percentage of active vs. sleep time - even if the sleep current is misread in such case, if you combine the duty cycle from one study with the carefully measured sleep current from another, you can form a good picture with only cheap tools.