The first thing is to make sure you aren't seeing copper from the jag - they usually color the patch if not wiped off. The same goes for bronze brushes.
You should not be surprised to see copper in a barrel that only sees a few rounds per year, especially in factory barrels. Custom barrels (Lilja, Shilen) will barely copper after broken in.
A heavily fouled barrel will require many patches and several hours to straighten out. You will remove alternating layers of copper and powder fouling in some bad cases.
If you are using a bronze brush, be sure it is clean - not just visually, clean it with solvent or get a new brush.
Apply Sweet's with one or two wet patches, let sit for 5 to 15 minutes, and swab out wtih a clean patch. Sometimes brushing with Sweet's is necessary - soak, brush, wipe out, and repeat if you need to.
Most guns will benefit from a cleaning with JB Bore Shine every two to three hundred rounds. This will get the fouling out that the solvents
leave. They also have a new product that is a finer Bore Polish to finish up with.
Put JB on a clean patch and push into the barrel. Short stroke just in front of the chamber to get the heavier fouling that is usually present at this location. Stroke the patch full length of the barrel several times, then push it out, run a solvent soaked patch to wash out the JB, and inspect for copper. Repeat if necessary. You can get aggressive with this abrasive, it won't hurt a factory barrel, and will probably improve it. Note that if you run a patch of JB through a clean barrel it will still come out black from the steel and the original gray color of the JB paste, so this is not an indication that powder fouling is still present - don't sweat the color.
If you get the barrel really clean it will shoot to a different point of impact, especially for the first shot from a clean, cold bore, so you probably want to shoot sighters before a hunt and clean little or not at all until the hunt is over.
I will bet your other rifles have copper build up also, it is just very hard to see in smaller calibers. Flashlights with an incadescent bulb will tend toward yellow color that reflects up the bore, either hiding the copper or making you think there is copper where none is present.
You might try one of the polishing kits with bullets coated in graded abrasives; this might accomplish two things - reduce the fouling since the barrel is smoother, and perhaps improve the rifle's accuracy assuming the ammunition, bedding, scope and so on are in good shape.
Good luck - you can get all the copper out, but it is difficult sometimes.