Ummm, I hate to be contrarian, but sometimes I can't keep my mouth shut.
First, neck sizing done correctly will improve accuracy. But the right dies are important. The Lee collet dies mentioned above can work as good as dies costing 5 times as much. I've only tried the collet dies once, and I had issues. After seating a bullet, the neck tension was so weak I could pull bullet out with my fingers. Not good. Kept adjusting the die per instructions trying to get more neck tension and it finally broke the die. Said fuggit and went back to my FL sizer.
Redding makes several different types of neck dies, some have worked and some have sucked in my experience. Their standard, plain Jane neck sizer die is a POS IMHO. Tried it with one of my cartridges and it induced terrible run out / concentricity issues. The FL sixer does a better job, when adjusted properly.
Their Type S and competition neck sizers with bushing is where they start to get good. With good brass and the expander ball removed, it will size necks straight. The interchangeable bushings allow you to fine tune neck tension and these dies also allow you to adjust how much of the neck is sized. I use a competition neck sizer for my 308 and only size about 2/3 of the neck. Using this sizer and a Forster micrometer seating die definitively improved accuracy on a rifle that was already very accurate.
One more note. Bad brass is just that. Bad. No matter what dies, press and technique you use, you can't fix chitty brass. If the brass thickness is not even around the neck, the only way to fix it is neck turning. Life is too short to waste time turning brass necks for a hunting rifle.