Soak the red onion in a small bowl of ice water, 10 minutes; drain and pat dry.
Meanwhile, put the potatoes in a large saucepan; cover with cold water and season with 1 teaspoon salt. Bring to a simmer and cook until tender, 10 to 12 minutes. Drain and transfer to a large bowl. Add 1/4 cup olive oil, 2 tablespoons vinegar, the sugar, 1/2 teaspoon salt and a few grinds of pepper; toss until the potatoes absorb most of the dressing.
Add the giardiniera, provolone, salami, artichoke hearts, olives and red onion to the bowl with the potatoes; toss. Let cool to room temperature.
Drizzle the potato salad with the remaining 3 tablespoons olive oil and 1 tablespoon vinegar. Add the basil and season with salt and pepper. Toss again just before serving.