Smoothing is great for on-screen presentation, but you need to be seeing what the print is actually going to produce.

There's a very complicated answer (!math!) for the question "How much should I raise it?", but the simplest explanation is that for MAXIMUM smoothness, no edge along a curve should be longer than 1/2 the step size of your printer (layer thickness or XY step size). Example: If your printer can do 0.1mm steps, then your edges should be no more than 0.05mm long (lookup "Nyquist Theorem")

I do realize that if you actually used that setting, you'd create nearly an infinite amount of polygons and the file size will be monstrous.. but you did ask "How can I get the smoothest print?" ?? You have to trade off file size for 'Accuracy around the curve'.

Examples:

If a cylinder is 0.5mm, I'll use 16 sides (because you basically can't see it ?? )

If a cylinder is 1.0mm, I'll use 32 sides

If a cylinder is 2.00mm, I'll use 64 sides

and so forth.. if you want a 200mm smooth cylinder, you'll be up at 1024 sides or more

Of course, the same thing applies to ANY curve in any axis, like a sphere. The more segments you have, the smoother your end result will be.