In a large saucepan, combine potatoes, 1 teaspoon salt, and enough water to cover. Bring to a boil over high heat and cook potatoes until easily pierced with a fork, 10 to 15 minutes. Drain potatoes and place them in a large bowl.
Assemble the salad:
In a small bowl, whisk together olive oil, vinegar, tarragon, remaining 1/2 teaspoon salt, and pepper. Pour olive-oil mixture over warm potatoes. Stir in celery, sweet and green onions, and pickle relish. Fold in eggs and mayonnaise. Cover and refrigerate for at least 2 hours or overnight.