The simplest approaches are the Lion Head Winter Route from the east, and the Ammonoosuc Ravine Trail from the west. The LHWR leaves the summer LH near the mouth of Tuckerman Ravine and climbs to the north edge of the ravine. When it first hits the side of the ravine it has one little (ten feet or fewer) near-vertical scramble followed by a short boulder-shaded section that's usually icy, and there's also a significant section of switchbacks that feels a bit exposed. Crampons advised, and some people suggest ice axes (I used mine once after a spring rain turned everything to ice). The Ammo trail feels safer - there's just one spot by the waterfall where you really don't want to slip. Both are heavily traveled, so expect hard-packed trails where some kind of traction will be helpful.
As you know, the main issue is not the trail but above-treeline conditions. Inevitably, both routes I mentioned keep you above treeline for a while. Winds are often from the west, so the LHWR might be more sheltered on its lower section, but somehow the Lion Head itself often feels windier than the summit.
How far down the southern presis are you planning to hike? If you've not been down the Webster Cliff trail before, be advised it's on the steep side for the end of a long day (not a scramble, just tiring). Crawford Path is much gentler (and has lots of parking).