In a theoretically ideal electric motor, adding an additional gear could increase top speed (by staying below the motor's RPM redline) but would not help with efficiency/range or acceleration. In fact if you gear it too high power output will fall.
In a theoretical motor you have a maximum torque and a maximum power limit. From zero RPM up to some specific RPM you have maximum torque available, and power climbs linearly with speed since power is torque x rpm.
From that RPM onwards you have maximum power up to the top speed of the car and torque falls linearly, again because power is torque x rpm. This is what an ideal motor would look like:
So if you're near the redline of the motor and you switch to a higher gear, as long as that higher gear keeps you in the flat peak power region you won't actually gain anything - the motor will spin slower but put out more torque - same total power, so same torque at the wheels for the same road speed after passing through the different gear ratio.
If you make the gearing too high you'll end up down the power curve (below 3500 here) and actually lose power output.
So apart from increasing top speed if it was previously limited by the motor RPM, adding a higher gear doesn't do you any good with this theoretical motor, you certainly don't gain any efficiency.
Adding a lower gear however can provide increased acceleration at lower speeds, which is why a very few very high performance EV's have two gears - like the Rimac concept one. They're not really gears as such since they vastly overlap each other - think of them more as "high range" and "low range" as you would get in some 4x4's or off road vehicles. You can do a wide overlapping range of speeds in the two different gears.
Real motors are not perfect though, many types of motors suffer from a modest fall off in horsepower and efficiency near their upper RPM limit due to issues such as "field weakening". This causes that flat red power line to start to droop above say 5000 in the graph above, and is part of what causes a real EV to start to run out of puff near it's top speed.
In theory an additional gear could bring that road speed back down the power curve to say 4000 where it is still at maximum and give slightly more power and efficiency. In practice, the additional power losses added by the more complex gearbox can exceed the small gain that the change in ratio would give. And those additional losses caused by the more complex gearbox are dragging performance down at
all speeds.
Not to mention you now have a gearbox that will wear out, as changing gears can never be wear free, whether it's meshing gears in a manual gearbox or friction clutches in some sort of auto box. So 9 times out of 10 a higher ratio for high speeds just isn't worth the hassle, and will reduce performance at slower speeds through increased losses in the gearbox.
There is also another approach - which is to design motors that come a lot closer to the theoretically ideal motor that maintains maximum power right to maximum RPM, and a lot of research is being done on that right now, and there are some examples of this already such as the motor in the BMW i3, and possibly the one in the Tesla Model 3, both of which aim to improve the performance of the motor near it's maximum RPM to minimise any droop in power output or loss of efficiency.
Electric motors are already pretty darn amazing, but there is still more that can be done to increase maximum operating RPM and maximise power output and efficiency at high RPM without compromising low RPM performance and efficiency, effectively widening the already very wide useful power band even more.
Once you have that there is really no need for multiple gear ratios at all, even in a very high performance car. As far as we know, the Roadster 2.0 is going to be a single gear ratio for example.
Rimac (both the old concept one, and the new not yet released model) is the only high performance road legal production EV I can think of which has two gear ratios.