I don't think there's a lot wrong with Redmans liners, they may not be the best, but they're "good enough", and capable of quite good accuracy. Here's a picture of a liner that I made to go into my .45-90 rolling block, and typical groups with it and iron sights at 100 yds:
(You need to Login or Register to view media files and links) Those groups were shot with Federal bulk pack ammo, and will consistently be under 1.5" at 100, and I have had a few as small as 5/8". Typical 90th percentile group is less than 1.25". This is not an fluke, I have had similar or better accuracy from several of these liners that I bought from Brownells.
I think the workmanship in installing the liner is critical to accuracy. John Taylor, as mentioned, does an excellent job. I like that he reams the bore to straighten it instead of just using a liner drill.
Curt Hardcastle, who replied to you earlier is meticulous and I'm betting would have an excellent result if he did a reline for you.
The typical gunsmith reline, where a 16" long 8mm piloted drill is used and then a liner shoved in and soldered or epoxied is good for restoring a can shooter, but isn't the best path to target grade accuracy. I've seen a lot of relines with a kink in the center, where drilling from both ends did not quite meet. As a minimum, I think that the entire installation should be drilled from the chamber end, and that an extended drill long enough to do the job is essential. I also think that it should be done on the lathe, with the barrel indicated in to the bore through the headstock. Unfortunately, that takes a longer bed than some people have to make it work. Like a previous poster stated, you pay for a better job, amazingly enough you usually get better accuracy.
For what it's worth, I think Beinke's break in thoughts are about right; 600 to 800 shots is what I've seen on new .22 barrels before they achieved best accuracy. That's with Lilja and Douglas barrels, too, not liners.
dave