Build a medium-hot fire on 1 side of a charcoal grill, or preheat a gas grill (if using a 2-burner grill, light 1 burner on high; if using a 3-burner grill, leave center burner off and heat other 2 burners on high).
Brush a rimless baking sheet with oil. Place fish, skin side down, on sheet. Rub flesh side of salmon with 3 Tbsp. oil and season generously with salt and pepper.
Brush grill rack with oil. Slide fish off sheet and onto grill rack over indirect heat. Cover; cook until salmon is just cooked through and just opaque in the center, about 20 minutes. (To test doneness, slide a paring knife into the center of the salmon, remove, and carefully press the knife to your lower lip. If it’s very warm, the fish is done.)
Holding baking sheet with oven mitts, slide under salmon; use another thin rimless baking sheet or a very large spatula to gently push fish onto first sheet. Transfer fish to a platter. Drizzle with oil; season with salt and pepper. Serve with lemon halves for squeezing over.