With your hands, remove the seeds from the tomatoes by squeezing them over a sieve, squeezing just enough to get the seeds out, then crush the tomatoes that you have in your hands coarsely and return them to the seedless liquid. Heat ¼ cup of the olive oil in a large, heavy skillet over medium-high heat. Whack the garlic with the flat side of a knife, scatter the cloves over the oil, and cook, shaking the pan, until lightly browned, about 2 minutes. Stir in the olives and cook until any liquid has evaporated and the olives begin to sizzle, about 1 minute. Add the tomatoes and oregano and bring to a boil. Lower the heat so the sauce is at a lively simmer and season lightly with pepper. Let simmer while you prepare the cod, spooning in a little hot water if the sauce begins to thicken.

Pat the salt-cod fillets dry with paper towels. Heat the remaining ¼ cup oil in a separate large skillet over medium heat. Add the fillets and cook, turning once, until lightly browned on both sides and opaque at the thickest point, about 10 minutes. Drain excess oil from the skillet and carefully pour in the tomato sauce. Bring to a quick boil, lower the heat so the sauce is simmering, and simmer 2 to 3 minutes, basting the fish with the sauce as it cooks. Taste the sauce and add salt and pepper, if necessary. Divide the sauce among four heated plates and top each pool of sauce with the fillets. Drizzle each fillet with additional olive oil, if you like.