Preheat grill to medium.
Mash minced garlic and 3/4 teaspoon salt on a cutting board with the side of a chef’s knife or a spoon until a paste forms. Transfer to a small bowl and stir in oil.
Check the salmon for pin bones and remove if necessary (see Tips). Measure out a piece of heavy-duty foil (or use a double layer of regular foil) large enough for the salmon fillet. Coat the foil with cooking spray. Place the salmon skin-side down on the foil and spread the garlic mixture all over it. Sprinkle with 1/3 cup basil. Overlap tomato slices on top and sprinkle with the remaining 1/4 teaspoon salt and pepper.
Transfer the salmon on the foil to the grill. Grill until the fish flakes easily, 10 to 12 minutes. Use two large spatulas to slide the salmon from the foil to a serving platter. Serve the salmon sprinkled with the remaining 1/4 cup basil.