Voltage drop in receptacle circuit - options?
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.