Fill a large stockpot with heavily salted water and the potatoes and bring to a boil. Cook until tender, 20 to 25 minutes. Drain the potatoes and quarter when cool enough to handle.
Mix together the relish, mustard, garlic powder, onion powder, smoked paprika, eggs, onion, bell pepper, 3 cups mayo and a pinch of salt and pepper in a large bowl. Add the potatoes and toss until combined. Add additional mayo if the salad looks dry. Sprinkle with smoked paprika. Serve immediately or cover and refrigerate until ready to eat.