The one thing that will definitely aid in accuracy by putting a custom barrel on a hunting rifle is going with a little heavier contour than a normal factory #2 or #3 sporter. I usually go with #5 or #6 and even Sendero contours on my hunting rifles. Lightweight sporter barrels have too much whip and can't dissipate heat as well.
The other benefit of a custom barrel is that the world is your oyster when it comes to deciding on a chambering. So long as you are an experienced handloader of course.
I do not hike while hunting with a round in the chamber, so I shoot groups with my hunting rifles equivalent to how many rounds the magazine will hold. Being that my 2 main hunting rifles are magnum calibers, they hold 3 rounds in the magazine and I judge their accuracy with 3 shot groups. Though i know from experience on playing around on paper that my rifles will hold excellent accuracy for 5 shots or more, it really makes no sense for me to shoot 5 shot groups with those rifles because that would be a very unlikely event in the field.
Answering your question regarding a more affordable barrel, it's really hard to beat a Shilen. Below are pictures of 3 shots at 100 yards with a new load I recently developed for my 6.5 Remington Magnum and a group at 300 yards with a load using a different powder. It is just a plain chrome moly barrel. I believe I paid somewhere in the ball park of $150 for the unturned blank when building the rifle. Cost another $40 to have it contoured for a total of $190. I have, or have had, Brux, Lilja, McGowan, X-Caliber, and Krieger barrels on other hunting rifles and yet in a hunting rifle platform, this cheap Shilen will easily match or best the accuracy of the more expensive barrels. I have 2 other Shilen barrels on varmint rifles that produce very tight groups as well.
The most impressive thing with this new load I developed using RL26 is that the first shot was from a completely clean cold bore. I normally foul my bore with 2 or 3 shots before testing loads, but I was curious to see what it would do. The second shot landed in the same hole, the third overlapped the other 2 shots. Extreme spread on the speed was only 23 fps from completely clean/cold bore to a slightly fouled and warmed barrel. Also notice there is absolutely zero vertical deviation in the group. Not bad for a low priced Shilen
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A group fired laying prone with a bipod from a small dirt hill 300 yards from the target using RL17 powder. This powder gives good speeds, but is known to prematurely burn out barrels so this is why I am working with other powders and trying to get away from it. Again, very impressive on a hunting rifle for a low budget Shilen barrel
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