Who makes a good short height hiking socks?

I still have my first pair of smartwool socks I bought 10 years ago.  I regularly wear Lorpen light hikers.

Smart wool. Darn tough. Fox river

I really like darn tough myself

You'll be good to go with the brands mentioned. I'd recommend wearing sock liners whatever you choose.

I typically prefer merino wool socks and when hiking summertime and they're all crew-sized socks.  For medium weight I like Darn Tough stocks and for lightweight, I like Smartwool.  REI also has their own brand which have done pretty well.  My non-merino wool socks I'm warming up to in the summer are Prosok bamboo socks.  They're a little thicker, but really wick away moisture and are quite comfortable.  

ROCK6

How do you guys like wool vs synthetic?

I don't use synthetics anymore or at least if they are they're a wool blend.  Some synthetics have some anti-foot-funk stuff built it, but most smell like rotten monkey butt after a day of hard hiking, foot sweat or wet from precipitation or creek crossings.  For me, it's either merino wool or bamboo.  I don't often use sock liners, but when I do, they're silk.

ROCK6

I have several pairs of Thorlos and they have been bullet proof.

If you're looking for a cheaper option, I know the Academy brand (Magellen?) makes a merino wool brand of sock.

I got a pair of darn tough "no show" socks.  A little warm, but otherwise comfy.

Smart wool.thorlos..but my Cabot and sons merino wool hikers are what normally gets worn.

All I own is Smartwool and Balston in several different thicknesses. Neither brand has ever let me down. I was lucky enough to find all 10-12 pairs I own at TJ Maxx and Marshall's for $4-$6 a pair.

Highest percentage of merino wool possible.  I have several different brands and would have a hard time recommending a specific, but I often get various wool socks from cabalas, sierra trading post, and even from Walmart on occasion.

Some of my personal favorites:

http://www.avantlink.com/click.php?tt=cl&pw=55963&ctc=j19end7nv300zk8a0104w&mi=11469&url=http%3A%2F%2Fwww.cabelas.com%2Fproduct%2FREALTREE-PACK-MERINO-WOOL-SOCKS%2F1769644.uts?searchPath=%2Fbrowse.cmd%3FcategoryId%3D734095080%26CQ_search%3Dwool%252Bsocks%26CQ_sort%3DpriceLow%26CQ_page%3D20
They often go on sale for $10 for two.  I probably wouldn't buy them at their normal price.  These are not heavy, but rather an average weight.  They are my go to about 75% of the time.

http://www.sierratradingpost.com/goodhew-expedition-socks-merino-wool-mid-calf-for-men-and-women~p~3971v/?filterString=goodhew%20socks&merch=search-zr-prod3971V#qaSection
These are really thick socks.  You would probably have trouble fitting these inside normal fitting footwear.  My heavy stiff sole hiking boots which I use when hauling large loads are probably a 1/2 size bigger fitting so I can run really thick cushy socks.  Also helps my toes from bottoming out when coming down steep hills with 50#+.  I really only wear these when on long hauling hunting trips.  Normal backpacking I use normal hiking boots.

https://www.walmart.com/ip/Rocky-Wool-Blend-Hiker-Medium-Men-s-Shoe-Size-6-ndash-9-2-Pack/17327676#about-item
I would rate these as my "beater" socks.  Obviously not as good as high content merino wool but I have had good luck with them.  I actually have two generations of that sock, some I got several years ago which are more like a heavy weight wool/ crew sock.  The current offering is closer to what its sold as, and that's a medium weight hiker.  The current or last version I got is a lower calf sock, but works perfect for average height hikers.  I also tend to use these as my "sleeping" socks.

IF you have $$ to spend you can't go wrong with Filson.

I could keep rambling on, but much like boots, socks are not one size fit all.  What works for me might not work very well for you.