Why does my Gama Silent Stalker .177 zeroed at 20 yds shoot 6" high at 30 yds?

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You may just be zeroed at a further distance. You could say in the photo it is zeroed at 10 and 60. But at 30 they are 2 inches high. Hope this helps.
 
I don't own, nor have I ever shot a Silent Stalker so I looked it up on the web. I did own a Gamo Whisper many years ago with a similar LDC built into the barrel that shot quite well considering the trigger it had. That kind of variance in height between 20 and 30 yards is extreme. I played with Chairgun Pro using typical settings and even tried a scope height setting much bigger than one would see on a conventional rifle and didn't come close to the variance you see in height.

I suggest you remove the scope and see what the iron sights look like at those ranges. I don't see the pellet as the cause but I am assuming the groups actually look like groups instead of scattered patterns. If groups still show that kind of variance in height consider returning the gun to the seller -- something is wrong.

 
"Slugging" is pushing a pell through the bore, feel for rough spots etc. Use a cleaning rod and look for clean rifling marks on the pellet.

Lapping is polishing the bore to remove any roughness, fire-lapping is fairly easy and can help improve accuracy. Hand lapping is more tedious. Both methods require polishing paste. I use fine valve lapping compound. You can google the procedure, or PM me.

A bad breach seal will cause inconsistency. This seals ur barrel to the compression chamber. Often it's a simple O ring, much easier to replace than the piston seal.

Have u cleaned the barrel? They often come filthy with oils/metal chips etc. right from the factory. The crown may be burred, as well.
 
Try this before tearing into your rifle. Keep moving farther away in 5 yd incrimates 35..40..45..etc. shooting 5 or so shots at each incrimate. Note how many inches high (above your original zero spot) each group is as you move farther away. Stop and take note of how far you are when you hit bullseye (hitting the same place when you zeroed at 20) again. If the rifle is hitting high at 30 physics will come into play and the pellet will start to drop off as you move farther away. Your second zero is the one you want to use for zeroing your rifle. Once you’ve found the second zero move back toward the target in 5 yd incrimates. Adjusting your sites with each incrimate and stop at the distance you want the rifle to be zeroed at.
 
Match pellets(Wadcutters) are designed for close target shooting for starters. Wadcutter pellets are often recommended for small game. They cut clean holes in paper targets for easy scoring. Also they cut larger wound channels and are as efficient as hollowpoints at close range. As long as the range at which they are used is held to about 25 yards or less, they're fine for hunting. At longer distances, they lose their stability and accuracy, being designed for 10-meter shooting at their best. 
Switch your pellet for a round nose type and zero in at 25m/27yards, This should put you on target at both ranges you mentioned and will keep your groups nice and tight.