next is the dial, they are held on a few ways depending on Model. They can be friction, which is just held on by a metal ring of some kind. those are actually far and few on American Pocket Watches.A few models of Howard,and early Rockford are really the only two that I remember off the top of my head. They rest are held in by tapered brass pins, which are usually the early versions. They are also held in by 3 or 4 screws. Most are 3. There are a few models that are 4. some Hamiltons are 4 post dials. Maybe others, I don't really remember what has what, I just know what to look for real quick.
This one has 3 posts, so the dust ring has to be removed, and these are usually missing for some reason. Then there are 3 small screws around the "rim"(?) that have to be loosened. The dial comes off gently. A very little prying, I use a small teflon or plastic wedge type thing that I found, for this. You can use a screw driver carefully separating the dial from the movement. Working in a circle. Then once the dial is off, retighten the 3 screws, just snug. That keeps them from getting lost while cleaning. There really is no reason to take those the whole way out.
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Dust ring and one of the dial screws in the pic above. The dial screw is the little screw, into the side.
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This is the balance assembly, and balance cock. I remove these next, this is one of the delicate areas of all watches. To try to sum this up...There is a hairspring stud, and the regulator pins that hold the hairspring. I might not have some terms correct, but I'll do my best here. The small spring(hairspring) has to be delt with carefully. The easiest way I found when taking a watch apart is to try to do what I can while it's in the watch. So, I look at the regulator(the part that looks like a little lever) this slides to lengthen or shorten the spring in a way, for adjustment. There are two pins, or a couple of versions on how this has to be undone. On this watch they are just two pins sticking down. Which means the Hairspring "should" just slide out.
Loosen the really small screw. That is holding the stud, then take out the screw holding the balance cock, or bridge looking thing!

To get that off? I use flattened brass, that almost looks like a screwdriver tip but wider, I also use a piece of wood at times, or even a plastic wedge. Just kind of working around the base lifting a fraction at a time until loose. If you bind it or force it too much? You can break the pivot on the balance staff or chip or crack the balance jewels. Or all. After the bridge or cock is loose and if the actual balance didn't fall out on it's own? Carefully push the stud through with a sharpened tooth pick or whatever you want to use. It should slide out of the regulator easily too. If not? You will have to carefully take a very small screw driver and separate the gap on the pins just a fraction. Usually they will slide right out. You kind of work all these things watching everything else. Just a little here and there and keeping in mind of the other parts along the way and checking to make sure nothing is being forced twisted or is binding to the point of destruction!
Added....The first picture above, If blown up, you will see the regulator pins and the stud screw..The screw that needs loosened is the horizontal screw, if looking at the movement flat! The small vertical screw, is not a screw! That is a minor adjustment and the hairspring stud and will destroy the spring if tried to turn to remove.