Hylander,

Several of my Savages have the same tendency that charlie b describes - to throw the one or two shots until the barrel temperature stabilizes.
Also, I find that a barrel that has just been cleaned tends also tends to need a couple of rounds to get the barrel stable.

I have learned to shoot a 'sighter or two' so the barrel can get the barrel prepped and to a working temperature.
On really cold days, it might take 3 or 4 rounds before the POIs get stable.

Once the barrel is stable, I get 5-round groups like your three rounds that are touching.
Once the barrel is stable, I have gotten 25-round groups within 0.6 inches.
I do watch the barrel temperature, so the barrel doesn't get too hot.
I use a 85 to 140 deg F barrel strip to monitor my barrel temperature on all of my target rifles.
Above 120 deg F, the bullets tend to drop about 0.25 inches, so the group size grows when I shoot the larger groups, or the POI moves for the 5-round groups.

I'd give the 'sighter prep' a try, just to see if it helps.
For hunting, all of those shots are effective, but for measuring, getting the barrel stable would probably give you better results.