It sounds as if the movement is slightly out of balance. If you look at the moving coil assembly at the base of the needle, you should see a 'cross arm' with three stubs (the 4th arm being the needle itself). The stub opposite the needle has a (normally coiled wire) counterweight on it to balance the weight of the needle. The two side arms may or may not have much smaller weights to provide any additional balancing of the movement needs at 90' to the needle.
Most accurate meters are designed to read correctly in the horizontal orientation. Depending on the quality of the fine balancing, they may show some offset when positioned in other orientations. Balance can shift over time due to handling, eg. a sharp knock can bend the needle slightly at the cross arm (where the leverage is highest). Hitting the end stop hard can do the same thing (hence the terms 'wrapping the needle' and 'burying the needle').
My advice is to leave the movement strictly alone. If you start messing with it, trying to bend the needle etc, you are likely to make it far worse, snap the needle, get ferrous particles in the magnetic gap and so on. It is sometimes possible to improve the balance by adding small blobs of varnish to the adjustment arms, but first you need to work out exactly where this is needed - is it just one side arm of one side arm, or a combination of side arm and main counterweight? etc. If you get it wrong, the linearity will go off (under-read at the left hand part of the scale, over-read on the right hand side - or the opposite). Add to this the risk of damaging or contaminating the hairspring or pivots and it's just not worth the risk unless you are a watchmaker.
Be happy that the movement works smoothly and use it horizontally.
P.S. In your case it sounds as if the counterweight ls a little heavy. Maybe the very tip of the needle is missing? Trying to reduce the size of the main counterweight can only end in tears!
EDIT: Out of interest, I checked one of the Simpson 260 interest sites. I noted that the 260 has a taut band suspension, so no pivots and hair springs, but it will still have the same static balancing arrangement and danger of dust entering the magnetic gap. In addition, there is the real danger of snapping the suspension band if you start 'tweaking' things. I also noted that the handle is designed to allow the meter to be operated in a sloping position. Does it read right when it is sloping rather than upright?