I don't think my advice is much of a departure from what's been said already, but a few things come to mind. (in no particular order)
1. Seems to be mostly fast ammo you're running there and for accuracy in a Savage, I'd be looking for standard velocity stuff. CCI Standard Velocity and Federal Gold Medal Target are both ones that shoot well in my guns. Don't hesitate to sample some match grade stuff if you see it on the shelf as the only option. CCI Green Tag comes to mind. I think you'll find CCI Standard Velocity will shoot well though.
2. What magnification did you have the scope set at? Assuming it's set at 8x. If it's set lower, it might be more difficult to be finding center target consistently, or one shot to the next.
3. I don't know what your experience level is, but I see lots of new shooters "pulling" the trigger. It's better to try and increase the pressure on the trigger slowly so that the gun doesn't move. You'd arrive on this conclusion yourself, but thought I'd throw it out there.
4. Breathing~some shooters hold their breath when they shoot, some only shoot on the exhale, some don't care either way.

Not saying which is best for you, but be aware of what you're doing, then mix it up and see if the results change.
5. Your "hold"~one thing I have to force myself to NOT do is lean on the stock with my cheek. Old habits die hard though. WHEN I do lean on it, I find my groups open up. When I gently have my face against it to get consistent eye relief (distance eye-to-scope) things go allot better.
6. Last...and maybe most important; tight groups are great, and a goal on it's own if you plan on being a target shooter only, and especially if you hope to compete. BUT, for a field gun...how important is it? I'd say you should be trying to get everything inside the 1" ring @ 50 yards if you plan on hunting with that rifle. Smaller groups, and small groups in general at greater distances req. a different grade of rifle, AND ammo. Guess my point is this~have some fun with the rifle, mix-up the types of targets you're shooting...try some offhand shooting...reactive targets...and so-on. All the advice in the world won't improve your results if you don't have a pile of trigger time with the gun. It doesn't need to be a stressful bench-shooting session trying to get tight groups either. If you want one-hole, 100 yard groups~take-up reloading with your .270. The .22lr is a versatile, fun caliber but it has it's limitations. For me, I consider it's useful range to be about 75 yards with standard velocity ammo. It starts dropping fast beyond that.
I could keep rambling...but I think that's enough rambling for now.

If different/better ammo (and some of these ideas in this thread) don't tighten-up your groups, there is a small chance there could be an issue (crown?) with the gun.