Heat the oil in a large skillet over medium-high heat. Add the garlic, ¼ teaspoon each salt and pepper, and as much spinach as will fit. Cook, tossing and adding more spinach when there is room, until tender, 2 to 3 minutes.
Meanwhile, heat broiler. Place the rolls split-side up on a foil-lined large broilerproof baking sheet. Dividing evenly, top with the ham, tomatoes, mozzarella, oregano, and ¼ teaspoon each salt and pepper. Broil until the cheese is melted, 2 to 3 minutes. Serve with the spinach.