First thing you need to figure out is how deep to seat the bullets. You are likely loading to a COAL which is not the thing to do. Get a bullet seating depth tool / gauge and figure out where the lands of your rifle are and then look at the bullet and case to see how close to the lands you can seat. I'd suggest 0.020" to 0.050" off the lands.
Then look at your sizing operation and make sure you are not over sizing the case.
You will also want to measure muzzle velocity to see how fast and how consistent the velocities are. This can be a function of the powder type and how you are metering it but it can also be a function of the load density (how much of the case is filled by the powder). A very low load density can cause wild velocity swings depending on how the powder sits in the case.
If you are currently using flat base bullets, stick with those. Boat-tails are way more difficult to get to shoot accurately.
His Tikka has a very deep throat, with lots of freebore. Seating depth likely won't change much.
OP, you say the Sako factory loads shoot .4 moa.
If you have any left, pull one of the bullets, to check its "form"and weight.
Look for a bullet with similar "form" of the same weight.
You could also weigh the powder charge. Some Sako factory ammo will have the type of powder they used for the commercial offering of the grade of ammo you purchased. It's a place to start.
When you shoot at your target, zero at 100yds.
Then you need to check velocities. If you don't have something to do this, you can extrapolate, by shooting specific distances, say 300 yds, and measure the drop. This will give you a measurment to calculate or go online and look at trajectory charts published by the manufacturer of your bullets. Some manuals will have these charts for specific bullet weights.
You will want to set up a "regulation load." This load will use a bullet of the same weight and form as the Sako, with a powder charge of your choice, that will duplicate the trajectory/velocity of the Sako round.
This will give you a point to "tweak" your loads.
1.5 inch groups with some cup and core bullets may be about as good as the bullets are capable of. Jacket thicknesses may not be as consistent as more expensive or better constructed bullets.
1.5 inch groups on your first attempt is pretty good IMHO. Good for you.
You may also want to check your loaded cartridges for run out. This can be an issue. Take your loaded cartridge, lay it on a very flat surface and roll it with your hand, while watching the bullet tip for wobble.
Tikka T3 rifles are generally tack drivers, or at least shoot acceptably well.
I would try increasing the powder load you posted by one full grain, don't touch anything else, and shoot again. Load "5" rounds.
Then load five more cartridges with 1/2 grain more powder, and shoot.
Do this with a box of cases, over 20 rounds. Tweak from the tightest group.
You don't mention your barrel profile, so allow your barrel to cool to ambient temperatures after shooting the cartridges in the magazine, which is usually 4 rounds for the T3.
Don't try to shoot groups from a hot barrel, then set your scope settings. Always set your scope for the game your shooting.
It sounds like the 308Win is your hunting rifle. When hunting, a game animal won't give you the opportunity to shoot until your barrel warms up.
One other thing to check out, is the parallax in your scope.
You can see this when looking through the scope at the target, then move your eye up and down. If the reticle is moving, you have a parallax issue. This will only take care of 1/4 - 1/2 inch of your group.