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I don't have a Socom but do have a STD M1A and I have done alot to my rifle like shimming the GC, Sadlak Oprod spring guide, Tin gas piston from Sadlak and reworked the stock by inletting it on the front end and shimming around the mag well and even with what I consider my best hanloads for this rifle I am only able to get 1.5" 2.5" avg with 40.0 of IMR 4895 and a Nosler 168 CC pill, you have to remember these are Service Rifles and yours is a CQC Rifle and if you are getting 2" groups at 100yrds that IMHO is pretty dam good
I noticed in your pics that your handguard seems to be in contact with the stock for a third or more of its length from the muzzle end. This is generally an accuracy no no. As mentioned above the bipod is a no no too. Bipods can put variable pressure on the barrell by transmitting variations in the shooter's hold.
plus 1 on these comments.

it's funny, but with the M14 platform, the littlest things make the biggest difference and usually if you are getting bad groups, it means it is more than one thing.

i used to get sub-standard groups and i shrank my groupings by almost 2" by doing these things in this order:

the handguard: if your handguard does not slide or wiggle a little loose or is pinned tight against your stock - you have a problem. you want breathing room between your handguard and forend. another similar problem is the tightness of your front band pressure. you should be able to move your barrel side to side slightly while holding the front of your stock stationary. another minimal area to look into that relates to your barrel, is that you have a scope mount gripped to your barrel which isn't bad, but anything clamped to your barrel is not going to help accuarcy. this is minimal compared to the other things i mentioned although i have heard those mounts are not very good. make sure your mount is sound.

the gas system: the main reason semi autos are not as accurate as bolt guns is the violent action of the blow back action. i shimmed my gas system (i.e. tightened the action if you will), which will bring more consistent shots. i have never had a M1A with the proper tightness in the gas system right out of the box. i would say a MUST for every M1A. gas lock should tighten at 5:30. i know you mentioned that you already "shimmed" but i believe there is also such a thing as too tight.

ammo: find the right ammo and dont settle on just one kind. i have never fired a 16" 14 but my guess would be that you probably dont need that heavy or hot a round. see how a 150gr does.

barrel cooling: im sorry if im throwing in basic 101 stuff, but o well. wait 5mins between each shot. especially with a 16" barrel shooting heavy loads.

cheek weld: this is the most basic but we all forget and have this problem. make sure your cheek weld is consistent.

stock: your stock looks sound, being that it is synthetic, but your trigger housing should have a snug lock-up. bedding will help for sure, but you shouldnt need that for basic accuracy.

IMHO, if you did all this and get 3MOA out of a 16" rifle with no magnification, you shouldnt ask for better.

by the way....i am totally digging your paint job on that thing. nice looking rifle. dont give up on that beauty. it's not an AR so dont expect results w/o showing it some love first.

oh. and ps.....get a gun vise and dont opt for the bipod shooting. bipods are nice but are not good for sighting in shots to test MOA>
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