01 - Place eggs in a saucepan and cover with cold water. Bring to a boil, cover, remove from heat, and let stand for 10 minutes.
02 - Drain eggs and cool thoroughly under cold running water. Peel eggs carefully to preserve the whites.
03 - Cut eggs in half lengthwise. Gently remove yolks and transfer them to a mixing bowl; reserve whites.
04 - Mash yolks with mayonnaise, Dijon mustard, vinegar or lemon juice, salt, pepper, and smoked paprika until creamy.
05 - Pipe or spoon the yolk mixture evenly into the hollow of each egg white half.
06 - Slice each olive in half; use one half for the spider's body. Cut the other half into thin strips for legs.
07 - Place one olive half atop each deviled egg. Arrange 3 to 4 olive strips on each side to form spider legs.
08 - Lightly dust with extra smoked paprika if desired; serve eggs chilled.