uCurrent Gold from EEVBlog is an excellent handy tool to measure mA or uA order of low current on battery powered IoT devices. https://www.eevblog.com/projects/ucurrent/
It normally run on CR2032 3v battery and it's voltage swing is capped to 0-1.2V. It also drains out poor quality cheap CR2032s rather quickly.
The web pages reads that you could extend the voltage swing by increasing the battery voltage, upto 5.5v. But don't be tempted add a DC jack to use external power supply!!
It's negative pole is used to make "virtual ground". It needs to be independent from the ground of DUT. So it's important the device runs on battery.