Hello Jack, I will start by saying just about every barrel currently produced by the small one/two man shops will shoot better than 95% of us here.
I am not brand loyal to any one maker, mostly it's about who has what I want, in what caliber and profile when I want it.
A truly custom barrel could take a year before you get ahold of it. That may be something you are willing to wait for, but perhaps not for your first rifle.
I personally own/have owned barrels from Green Mountain, Colerain, Rice and Oregon. They are all more accurate than me.
If you are "cost sensitive" then the Green Mountain (in straight barrels - $150'ish) or the Colerain (in straight, $160'ish, swamped, $190'ish or a their single tapered profile, $175'ish) may be the ticket and "most times" can be bought out of existing stock from one of the various suppliers.
Oregon barrels are by order only. They make straight or tapered barrels but they are totally custom - you pick the size, length, caliber, twist, rifling depth, breech thread size/pitch but despite that you can usually get a barrel in about 90 days - they sell at a premium to the above two I mentioned, but are a fine barrel if you want/need all the choices these barrels offer.
An Oregon barrel goes for $200/straight, $225/tapered.
Rice barrels are the most expensive of all the (prolific) builders. They have a good following which tends to command a premium if the rifle is later sold. Depending on what exactly you are after you may have a wait time of 6-12 months for a barrel if it's not "in stock" somewhere. Certain profiles are not (cut) until he has 10 orders (justifies changing the equipment around).
Rice barrels range in price $200/straight, $265-305/swamped and $250/tapered.
Green Mountain uses a different barrel steel (1137) than most of the others (12L14), but unless that means anything to you that is a whole other discussion best avoided, at least in this thread.
In general Rice and Colerain use round bottomed rifling (supposed to clean and load easier - but Rice will also cut square rifling "regularly" in a few barrels when he does a run) and GM and Oregon use square (supposed to be more accurate) - so can toss a coin on that one.
Other barrels you may come across of which I have insufficient first hand knowledge to comment on include:
Long Hammock, Ed Rayl, Getz, FCI (Charlie Burton) and I'm sure I'm missing one or two contemporary makers.
Any of them will probably more than suit your needs. Once you finalize what profile/length of barrel you want that may help you make your decision.
I like all the ones I shoot and wouldn't try to steer you to one over the other - they all shoot "just fine".
(just noticed your location - you are about 1/2 hour from Rice barrels = Mocksville, NC - might be worth a call/drive down - he sometimes has barrels laying around)