Place the potatoes into a large pot and cover with water. Bring to a boil over high heat, then reduce heat to medium-low, cover, and simmer until tender, about 20 minutes. Drain and allow to steam dry for a minute or two.
In a bowl, whisk together the dry Italian dressing mix, tarragon vinegar, water, and olive oil until thoroughly blended. Set aside.
Place the hot potatoes into a large bowl, and roughly but thoroughly slice them with a table knife until the potatoes are in chunks. Pour the dressing mixture over the hot potatoes, toss to coat, and let the potatoes cool. Add the celery, bacon bits, dill pickle, and green onion to the potatoes; lightly stir in mayonnaise until all ingredients are well combined, and serve.