What you want to do is saturate the transistor, which means you want a base current (very approximately) 0.1* the collector current - around 0.3mA, or higher currents won't hurt. Assuming you're switching from a 3.3V MCU output, you'll have about 2.6V across the base resistor after subtracting Vbe, so Rb should be less than 2.6/0.3 = 8.67kilohms (choose 8k2 or even reduce to 4k7).
What voltage does that give between collector and emitter (Vce)? Check the datasheet for your chosen transistor; it'll probably guarantee "below 0.2V". e.g. for the BC847, Table 7 shows "VCEsat : IC=10mA; IB=0.5mA 90(typ) 200(mak) mV"
So Vce would be under 0.1V at currents below 10mA - and note the base current is only 0.05* the collector current.
Also see the graph in Figure 3 which shows measured performance of a sample transistor, where Vce (at 25C) is only around 50mv under these conditions, rising to 100mv at 30mA.
If that's not enough, the higher gain BC847 reduces Vce to about 30mv under the same conditions (see Fig.11)
Most small signal NPN transistors should have similar info in the datasheets, I've just used the BC847 (aka BC107 for old timers) as an example.