Not sure, but it seems the current model has a 1:10 twist barrel. If yours is the same, you should investigate shooting the 150 gr Match Grade VLD Hunting bullets from Berger. The Hornady 150gr IB has a similar BC. The Sierra Game king is trailing quite a long way behind either of those. There is no substitute for shooting the highest BC bullets in precision hand loads that you make yourself in once fired brass.
Getting your rifle to group better will take potentially working on the bedding, barrel free floating, and the trigger. The first 2 you should have no problem doing on your own, the objective being to prevent any intermittent loads on the barrel and the best possible (and most repeatable) coupling between action and stock. Youtube has plenty of videos on the subject or you could talk to your local gunsmith. If you have a wooden stock, have it pillar bedded, that way the expansion and contraction of the wooden stock no longer impacts the tension of the 2 action screws.
With the trigger, unless you know exactly what you are doing (and have the "right" remington trigger), I suggest you solve this problem externally. The most modest cost solution is to get a Shilen trigger. They offer this trigger as part of their custom rifles
so it is a pretty good trigger and much better than the standard Remington one. It is adjustable, but I have not touched mine. The only negative thing I have to say about mine is that they EDM the entire trigger profile and that leaves a square edge on the part that touches your finger. It would be much nicer with a rounded edge, but that should be done before the trigger is finished or assembled, because I believe it is through hardened so very tough and if one grinds on it with a dremel it would need some sort of re-finish to prevent corrosion. I hope someone from Shilen reads this, since otherwise it is a great trigger and modestly priced compared to some of the alternatives. There are gunsmiths charging $100+ to stone your trigger sears and adjust it and that is without the cost of shipping your barreled action 2 ways !
Remember with the Remington, that the safety never in fact blocks the striker from falling. Always use respect and follow basic firearm safety rules when handling a rifle with a loaded chamber. Use serious discretion about what kind of trigger pull weight to use on a hunting weapon.
Besides the above, how do you handle the shooting position in the field ? Do you use a Bipod, hunt from a seated position or prone ? I just recently got one of the long Harris Bipods which can support the rifle at an acceptable height when seated on the ground. A decent rifle rest and to be able to aim stress free is a big factor in making shots beyond 300 yards.