Build a medium-hot fire in a charcoal grill, or heat a gas grill to high. Rub corn with 1 Tbsp. oil. Grill, turning frequently, until corn is charred and heated, 10-12 minutes. Remove from grill; when cool enough to handle, cut kernels from cobs and transfer to a large bowl. (Corn can be grilled 3 hours ahead. Let stand at room temperature.)
Place onion in a strainer and rinse with cold water to mellow its flavor. Drain well. Mix onion, remaining 5 Tbsp. oil, tomatoes, basil, 1/3 cup lime juice, and thyme into corn. Season to taste with salt, pepper, and more lime juice. (Salad can be assembled 1 hour ahead. Let stand at room temperature.)