I just corrected the same issue on my GXV140 . . .had done that for years, but then started a thin constant haze.Tore it down and did a light hone (negligible cylinder wear) and rings, and like new again. There are three factors in play here: 1) Wear 2) Honda used a 1 piece cast iron oil ring in early engines that just didn't work or hold up that well, and has replaced with a 3 piece set. 3) On mine, all 3 rings had rotated to have the gap at the bottom, so oil could get into the cylinder at rest. This was a head scratcher for me, since the longer it sat unused, the more smoke at start . . .
If you do go in, plan rings, upper and lower oil seal, and a few gaskets . . . a couple hours and not much $$$ at all.
Oh, and one more thought (and not sure it applies to the 120 at all). On the GXV140, part way through its life, along with changing the oil ring style, Honda aso added a stem seal to the intake valve. The head didn't change, so this would also apply to older engines (It goes over the stem, and is retained by the valve spring thatbgoes over it). Since the GXV120 may well have an almost identical head, this may work on yours as well. It was $3.85 US as I recall, and certainly can't hurt. These changes by Honda, though, tells me thqt they knew that they had an oil problem in the earlier GXV engines . . .