My gun (like your black one) has a 10 X scope. It can bullseye a quarter from 50 yards with cheap ammo and do it all day long. Buy good ammo and the groups shrink to the size of a dime.
(Shooting from a sandbag rest and a sturdy bench.)
If your scope monts are out of alignment the gun will do weird things. You will sight it in and you will think it's all right on the money but it will just go out of whack on you for no appearent reason.
I spent a whole day sighting in the gun, only to have it go off again, five minutes later.
Another symptom is if you can't get the gun to shoot on the paper, no matter how far you turn the adjusters.
If your gun does these things, it's probably best to take the scope off and remount from scratch.
Also, on cheaper scopes, it is possible that they won't hold their adjustments very well.
Some of the more "budget priced" scopes are fine for plinking and field shooting but remember the old saying: "Good, fast or cheap... Pick two out of three."
I'm not the greatest expert in the world but here's what I did to get mine shooting well.
1) New Barrel, new stock & reworked the trigger. (Kept same trigger but polished the hammer/sear.)
2) Rimfire Technologies blueprinted bolt, guide rod, spring and bolt buffer.
3) Sightron 3-10X scope.
To install the scope:
1) Ensure the windage/elevation adjustments on the scope are set to exact center.
I got a couple of pieces of wood to lay on the table and rolled the scope across them while looking through the eyepiece. If the crosshairs are not centered, they will appear to wander as the scope rolls. Adjust a click or two and repeat test until you are satisfied the scope is at center.
2) Install scope on rings but tighten only to snug. Ensure everything is mounted plumb, level and square.
3) Shine a light out the barrel. I use one of those battery powered bore inspection lights.
http://secure.armorholdings.com/kleen-bore/product109.html
If you want, you can just use a regular flashlight and a dentist mirror. Whatever method you use, the goal is to make a spot of light shine on the wall when you point the gun.
4) Go into a dark room with your gun and light. Aim the gun at a wall, a fair distance away.
I did this in the basement where I can get a good 25 foot shot from end to end.
If you have a gun vise, this will be easier. If not just put a sandbag or a pillow on the table.
5) Look through the scope and see if you can find the dot of light on the wall, coming from the barrel of the gun.
Use the adjustments in the mounts to bring the spot to the exact center of the cross hairs.
Tighten one side of the mount base or loosen the other (alternately) until the cross hair moves onto the center of the light spot.
You can use shims in the bottoms of the rings to alter the vertical alignment but, if you have been careful when you did the mounting in the first place, this should not be needed.
6) When you get everything lined up the way you want, tighten all the screws down to the recommended torque. (Approx. 20 inch-pounds.)
7) You are ready to take the gun out to the range and sight it in.
Again, this is the way I did it. Other people use bore sighters to align the scope. Other people use those laser thingies. Personally, I do it with a beam of light going down the barrel. If that's how light goes down the barrel, so will a bullet. Right?
You already know how to bed your action and tune the rest of the gun. That should be academic for you.
But, in my experience, if you don't get that scope mounted on the gun, straight and true, there's no amount of tinkering that will make it right.
Thing is, when I got my Weigand mount, all that went out the window. I carefully bolted the mount onto the gun and installed the scope in the mount. It was perfect. No fussing around.
And, if it gets bumped or dropped, there's no worry about knocking it out of alignment because there's nothing in the mount to move out of alignment. If you hit it hard to move the scope, you probably broke the scope anyway.
I used to spend a lot of time fussing with scope mounts. With the Weigand mount, I hardly ever worry about it.