Re-ringing without at least "Deglazing" the cylinder is just throwing your ring set money away. If your cylinder has any vertical scratches in it from the old rings or piston scuffing the new rings will not seal against them, and there will be little of the oil controlling cross hatching left. This will cause the new rings to run dryer than the old ones, and put more strain on the rod. The new rings will likely wear out even faster than the old rings that are in it now, as well as scratch the block even more deeply. The best you can hope for is that the new rings would have less end gap resulting in less blow by through them. And since you'll have to drop the pan to remove the rod cap to remove the piston so you can hone, (Does Amsoil sell cutting oil?)

might as well do it all. Check your valve guides while you have the head off. There's the slight chance that may be where your oil is entering the cylinder. And at least hand lap the valves before you reinstall them. Do all your work, especially measurements, with all the parts at room temperature to eliminate thermal expansion from the equation.
I know this sounds like a lot of extra work, but the end result will be so much better and long lasting.