The following are a few steps to help increase the accuracy of these wonderful single shot rifles:
First thing to increase accuracy is floating the barrel.
This is done in two steps. First step is to go to Home Depot or Auto Zone and get a flat piece of rubber material. It only has to be 1/16" to 1/8" thick. Cut a hole in the rubber large enough to slide over the forend hanger lug. This will fit between the barrel and the forend. Just snug it up, don't crank down on it.
Second step (this is optional) is to sand some of the stock material out of the forend to make sure it doesn't touch the barrel. I like to put two business cards together and be able to slide them between the stock and barrel without obstruction.
Second thing is trigger job.
There are many homemade recipes for reducing trigger pull, but the safest and best way is send it H&R 1871 Inc. They will bring the pull down to 3 pounds free of charge. You pay shipping though. For a small extra fee, they will reduce it lower that 3 pounds. Another option is to take the rifle to a qualified gunsmith. The fee is usually $25 to $40. Being that the NEF has a solid, one-piece receiver, it makes assembly difficult, it can be done though with much patience and loss of hair.
Third thing is bed the scope mount.
This is optional, but you definitely need to take the screws off and put Loc-Tite on the screws and tighten them down real good.
To bed the scope mount, remove the mount from the barrel. Also take the scope and rings off the rail. Drill two small holes in the end of the mount that hangs over the end of the chamber. Don't drill all the through. Just make them shallow holes. Also rough up the same are with some 60 grit sand paper. This gives more surface area for the epoxy to grab. Now spray the top of the barrel at the chamber area with Pam cooking spray. This is the release agent. Clean the bottom of the rail with acetone or gun scrubber, it has to be squeaky clean for the epoxy to stick. Now mount the scope rail and screw it down tight. Mix up the epoxy (I use JB Weld) and press it down into the space between the barrel and rail. Some tape on the sides will help hold the epoxy in. Let it set up for an hour or so, then clean the excess with a popsicle stick and some acetone. Now let it set up over night. Next day you can take it off and clean it up with a file and sandpaper.
Fourth thing is location of sandbags.
When shooting from the bench, make sure that your rest is moved as far back as possible. This will make things more consistent from shot to shot.
This should get you started. I suggest before doing any of these modifications, take the rifle to the range and put 30-40 rounds through it and see how it shoots. It may shoot good right out of the box.