Place the potatoes in a large pot; cover with water by 2 inches. Season generously with salt and bring to a boil. Reduce to a simmer and cook until the potatoes are tender, about 20 minutes. Drain the potatoes; let cool 15 to 20 minutes.
Whisk the olive oil, vinegar, 1 teaspoon salt and a few grinds of pepper in a medium bowl; stir in the olives and diced celery. Halve the potatoes and transfer to a shallow bowl. Scatter the tomatoes over the potatoes, then drizzle with the vinaigrette. Scatter the celery leaves, basil and chives over the salad and season with salt and pepper.