From Grand Paradis (1056m) you can hike to the Susanfe refuge (2102m) where I suggest to stay for the night. The trail to the refuge leads over the Alp Bonaveau (1550m) and Pas d'Encel (1956m). Up till there it's easy hiking with some exposed parts around Pas d'Encel which are secured by chain ropes.
From the Susanfe refuge it's an hour's hike to Col de Susanfe (2495m), the start of the Haute Cime's south ridge. You can also reach the Col de Susanfe from another high valley, the Salanfe.
At the Col the trail becomes very steep (30-45°) and winds east of the south ridge to avoid a few vertical rocks. At approx. 2850m the trail is back on the rifge where it stays. At approx 3050m you reach the Col des Paresseux where you get the first look at the last 200m of the summit. From here on the slope reaches 50-60°, you will need your hands or your ice axe.
You definitely need an ice axe and crampons if you climb the Haute Cime's south ridge before July or after August. Due to the steep summit a few ice screws or other securing material might also be useful.
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