The answer is yes, you can mitigate voltage drop by running a larger wire to the first outlet, then normal size wire from there. Voltage drop is additive.
Let's say you run 10 gauge 100' to the first outlet and 12 gauge 60 feet to the last outlet, and you have the full load of 20 amps on a duplex receptacle at that last outlet. (Very unlikely but just as an example.)
You'd drop 4 volts on the #10 to the first outlet, then another 3.8 volts on the #12 to the last outlet, for 112.2V - that would be marginal, 6.5% voltage drop. (Just used an online calculator for these calcs, but they seem about right.)
I am getting #6 wire to keep 120V at full load of 20 amps under the NEC recommended 3% voltage drop. Nobody would do that but that's the actual worst case. In reality it's pretty rare to see voltage drop problems with #12 wire in residential systems.
Is it important to keep the voltage drop to 3%? Would 5% be OK? What's the limit?
Depends on what the utility delivers, it may be under or over 120V. Lots of things run fine on a wide range of voltage, others are very fussy about voltage.
Will the load really be 20 amps? Maybe every morning with a space heater and a hair dryer running, more likely well under that. Could you get by with a 15A circuit? What will the actual load be?
Can you route the cables differently to shorten the run? Could you run it more staight line to the first outlet through the basement or attic, or maybe just bore holes higher in the walls to miss the doors?
If you really want to keep voltage drop really tight, your might have to consider rearranging things with a subpanel at the other end of the house. That will take some effort and expense, and you'll have to find a spot for the subpanel, but you may offset that with savings on wire and long convoluted pulls, and it will free up some space in your main panel.
Straight up
You cannot afford to fall below 110V. Ok.
Supply at your house is supposed to be 120V so we have a little room to play with.
If you use 8 AWG cable, voltage drop will be limited to 8.29V on a full 20A load, giving 111.71V. That meets your criteria. If actual pulled amperage is less, voltage drop will be proportionately less, and voltage will be higher.
If that is not good enough, you can run thicker wire. Any bigger than #8 and you should be looking at aluminum wire, e.g. a #1 Al cable will create a 1.46V drop, giving 118.54V.
Transformer
Now if you power the circuit off a 240V breaker, you can do your transmission at 240V and power a 240->120V transformer. A 5 KVA transformer will more than suffice, and is often available on Craigslist for as little as $100.
Nominally you assume 2:1 on the transformer, but you can jumper the transformer for slightly more or less if transmission voltage drop warrants it.
Anyway, since your feed voltage is 240V, you have half the voltage drop and 1/4 the power loss.
14 AWG wire will suffice. At a full 20A on the 120V side (10A at 240V), it will drop 16.08 volts off 240V (8.04V on 120), giving 111.96 volts at your plug.
Two transformers
You can use 2 transformers back to back to kick the transmission voltage up to 480V, 575V or even 600V, and voltage drop will be considerably less still. However, we can't use wire smaller than #14, and #14 wire is getting the job done as above. So this is a bad idea given the higher hazard in the higher voltage section.
Wait. I thought 110 was normal, I meant no drop at all
Then install an "always online" type UPS at the shed. The online UPS always fabricates AC power through its onboard inverter, so it's always making it to the spec you paid for. The supply side of this online UPS will have an input tolerance; wire thick enough wires to satisfy that, and if at all possible, use 240V for thinner, cheaper wires.
But I wanted to compensate with transformer taps
Doesn't work. You can't hold voltage steady by selecting a single transformer tap, because voltage drop is proportional to actual load. Whatever tap you select, voltage will go up and down with load, you're only affecting the range bracket. You can get transformer-based equipment that will automatically "select-a-tap" as needed to keep voltage within spec, it will step up to the next tap as voltage sags etc. But this is not perfect, there will be some sawtoothing.
Best Answer
Volt meters are rated in mega ohms per volt. Most meters today are at least 1M ohm with many being 10M this so the meter won't load the circuit and provide an inaccurate reading. Since the impedance of the meter is so high it can detect voltage with almost no potential or what we call phantom voltage. There are low Z or low impedance meters out there that put enough of a load that phantom voltages won't be read, but in electronics trouble shooting low Z meters affect the circuit that's why most meters are high impedance.