If the back of your watch is completely smooth, the odds are that it has a snap-fit case. Screw-back wristwatch cases generally have small indentations in the case back into which you fit a case wrench. Rolexes have an elaborately ridged edge to their case backs and take a special wrench. A few early water-resistant watches have little screws that hold the front and back of the case together, but these are usually quite obvious, and you didn't mention them in your description.
Most "fancy" ladies watches of the 40s, 50s, and 60s, had snap-back cases. To open one of these without damaging the case, I recommend gently working a decent quality case knife into the crack where the case back joins the case and gently levering up. If the case is really snug, you may need to do this in a few places to gradually pry apart the case halves. Often, the best place to do this is at the lugs, because they offer a leveraging point. I don't recommend doing this with a sharp knife blade, as the resistance from the case will probably damage the knife edge, and if the knife slips, it can easily scratch the case. If you don't have a case knife, try using the rounded edge of a butter or table knife. Do not use a flathead screwdriver blade, as the short width of that tool can easily place so much pressure on a small section of the case that it deforms and can never be made to look good again.
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