Place the eggs in one layer in a large saucepan. Cover with cold water by at least 1 inch (2.5 cm). Add 1 Tbsp vinegar and 1 Tbsp sea salt. Bring to a rolling boil over high heat and cook for 10 minutes.
Transfer the eggs immediately to very cold water or an ice bath. Cool for about 10 minutes, then peel.
Cut the eggs lengthwise and place the yolks in a mixing bowl. Add mayonnaise, Dijon mustard, apple-cider vinegar, paprika, garlic powder, sea salt, and black pepper. Mash with a fork until creamy.
Pipe or spoon the filling into the 12 egg-white halves.
Cut 6 olives in half lengthwise and place one half, cut-side down, on each egg as the spider body.
Cut the remaining 12 olives in half lengthwise, then slice each half into 4 strips. Arrange 4 strips on each side of every olive body.
Cover and refrigerate at 40°F (4°C) or colder. Serve cold.