If I make a complete guote (one that ends with a period), and then make an addition to add to that thought, does the second quote begin with an em dash? "I want you to pick up milk, eggs, bread, and something for dinner." "Okay" "--also, get something for tommorow!"
No. Treat it as a new sentence. "I want you to pick up milk, eggs, bread, and something for dinner." "Okay" "Also, get something for tommorow."
