This is called the Centre Axis Relock, or CAR system.
Woodrugh still shoots from a conventional firing position, but his holstered/resting position is with his gun pulled close to his chest, barrel pointed out at the expected direction of fire. Whilst holding a weapon at arm's range can increase aim and stability if using a high calibre weapon (especially absorbing recoil), this grip is considered better in small spaces with low calibre weapons.
Not only does it create a stance from which it is possible to present a smaller target area to any assailants, it also allows the holder to fire from cover whilst still moving without too much adjustment.
It's also an effective way to carry a weapon if wearing full combat gear, which given that Woodrugh was a former soldier would also make sense as to why he is more comfortable with it.
The main advantages of this stance are increased weapon retention. It's much more difficult to take someone by surprise and force them to drop their weapon (or even disarm them) if their weapon is held too tightly to their chest. As most people (seasoned vet's included) reflex reaction to a loud noise (such as a gunshot) is to flinch and draw the arms in, in a combat situation that movement can cost valuable seconds: the CAR system eliminates that reaction, as the arms have nowhere to move back to.