Bigger accuracy gains are probably available to you by testing several different kinds of ammo, and finding which one works best in your rifle, if that is an option available to you.
Once you have selected the ammo that works best in your rifle, you might (or might not?) want to try further improving the performance you get from it. Sorting by weight, rim thickness, etc, will never hurt your accuracy, and might conceivably/possibly help it. At worse, it'll be useless (i.e. the only thing you have to lose is your time and effort). Be warned though that small improvements in accuracy are notoriously difficult to detect and to nail down, by the time you eliminate false positives and false negatives.
W.r.t. ammo choice... .22s are notoriously picky and unpredictable. Shooting match ammo is a good thing to do, but there are no guarantees that by using good $10/box ammo you'll get good results in your rifle.
About 15 years ago I bought a couple of cases of .22 match ammo, for myself and a friend. The ammo seller had an indoor 50m range with a machine rest, and if you were to buy a case of ammo he would test your rifle (barrelled action actually) to let you pick the ammo that works best in your rifle. So I got to test my rifle firing 10-shot groups from a machine rest, with different kinds of ammo, in absolutely zero wind conditions. And a very pleasant couple of hours with a wonderful older shooter.
My rifle was an Anschutz 54, and we shot about 6 or 8 different kinds of ammo - for the most part, the "budget match" and "super premium match" offering from each of the major manufacturers. From memory we fired:
RWS Target
RWS R50
Eley Target
Eley Tenex
Lapua Dominator
Lapua Biathlon
It was surprising how unpredictable the results were. For example, RWS R50, which is used by shooters to win the Olympics, was truly mediocre in my rifle (and the RWS Target was pretty mediocre too). It's not that RWS is bad .22 ammo, far from it - but for whatever reason, it simply did not shoot well in my rifle.
It ended up that the two kinds of Eley shot best in my rifle, almost equally well in fact (the bottom-of-the-line, and the top-of-the-line offering), so I bought a case of each.