Years ago, a friend and colleague of mine who was a competition shooter, improved my handgun shooting literally over night! He had me to concentrate my focus on the "front" sight, and the "target" only... The rear sight, although obviously also in the equation was only a blur, and with just a bit of practice, I improved my accuracy with off-hand shooting by about 100%...
Now, as far as lever action rifles (or any other kind for that matter) I still claim not to be able to shoot that well at a distance, still can not see the open sights well enough to hunt with them, and have gone against the "traditional" view and have mounted scopes on my hunting lever guns. I.E., my 1895SS Marlin .45-70, and my Marlin 336 .30-30... I realize that many folks feel that scopes and slings on these guns are inappropriate, but since I simply can not hit with them at any distance without a scope, and since I can drive nails with when WITH a scope, I've bit the bullet so to speak, and use scopes on them...
But try that focus thing on just the bull and the front sight, with only secondary awareness on the rear sight, practice with it a bit, and I'll bet you find, as did I, that it does work... Just IMHO... Dave