you can take them apart ..
barrel length makes a difference to sound as does ammo .... Also, this may sound silly, but if your 1m away from a rifle, it's much quieter than holding it, when it goes off... Lol
Get someone o fire it, while you stand bout 20' away & see how it sounds, then you will get a better idea of how the rabbits hear it
They are normally very accurate rifles.
You need to "split the problem". Take the mod off and see if accuracy improves. If it does then the mod or the mod fitting is faulty. Try someone else's mod if possible and do some logical fault finding.
Do this before you start stripping the mod. It's easy to make problems by diving into something rather than solve the original problem.
Is the rifle new or second hand ?.
if its new just keep useing it may go a little quieter with some carbon build up,but to be honest it shouldnt be loud for a 22 dont bother stripping it down you wont learn anything ,the mod i mean (sak)
as others said check the scope mounts ,check the mods fitting good,let someone else have a go,you just said that youve had a new scope fitted i know thats not the noise problem but sounds like it may be the accuracy,ps my saks quieter than my parker hale on my 22
i have a 452 varmint cut down to 14 1/2 inches and a SAK. I have to say, I can't see any reason to change it after several years. Try looking at the variables first, as suggested,because I'm not sure that you will get much more performance on the noise front from a different setup. Accuracy wise, it would be worth checking thread, crown, scope alignment etc.
As just mentioned i would check the Thread on the barrel a friend of mine bought a secondhand CZ 452 and accuracy was all over the place and even worse with a moderator on, to the naked eye it looked ok but he took it to a gunsmith who checked it and it was slightly out he had the barrel shortened and thread re cut and now its spot on.
I use a Parker Hale mod on mine and i find it just as quiet as my Sak.
Edit - Just noticed your in Suffolk if your anywhere near Framlingham take it Jason at Trulock & Harris for a quick look.
dan71 - So far there has been some good advice given to try to help you.
Just to add my little bit in the hope that it might help:
Have you had the threading and crown checked (After checking the scope and mountings) if not then that is where I would start and if all is fine with that I would then try it with maybe a friends moderator or get a friend to try their rifle using your moderator - It is a "slim" possibility that the SAK moderator you have is not quite spot on - Maybe the baffles are starting to burn out - Often the case with pre-owned SAK moderators if they have been used with a .17HMR that has had the barrel shortened!
Following on from that, if the threads and crown are fine then it might well pay to try a few different types of ammunition to find which your rifle prefers: I am sure that you are aware of this already (And I don't want to sound patronising or anything like that as I don't know how much experience you have with rim-fire rifles but you have not mentioned what type of ammunition you are using nor have you mentioned if you have tried different types of ammunition) but it is a fact, particularly with rim-fires that different rifles behave differently with different types of ammunition so you need to find which type of ammunition your rifle "prefers". Plus of course if you are by chance using HV ammunition in stead of sub-sonics you will certainly find a big difference in noise and quite likely the same big difference with accuracy. (My Anschutz 525 loves either Magtec Sub-Sonics and Remington Sub-Sonics but will not give acceptable accuracy with most other types of Sub-Sonic ammunition - and it truly hates any type of High Velocity ammunition)