Remove giblets from chicken; discard. Rinse chicken inside and out; pat dry. Cut off and discard excess skin and fat around body cavity.

In a small bowl, mix oil, garlic, thyme, 1/4 teaspoon salt, and 1/4 teaspoon pepper. With your fingers, gently loosen skin over breast, being careful not to tear skin. Gently rub thyme mixture evenly over breast meat beneath skin.

Prepare a charcoal or gas grill for indirect heat (the charcoal or gas flame is balanced on the sides, not directly under the cooking area).

Rinse can and shake dry. Pour wine into can, then carefully lower chicken over can until can is inside body cavity. Gently pull legs forward so chicken stands upright on can.

Carefully set chicken and can, in this upright position, on center of grill (not over direct heat). Cover charcoal grill and open vents; close lid on gas grill. Cook until an instant-read meat thermometer inserted through thickest part of breast to bone reaches 160°.

Using oven mitts, carefully remove chicken with can from grill, keeping it upright. Let chicken stand on can for 15 minutes; then, holding a towel around the can to steady it and protect your hands, with a large carving fork, carefully remove chicken from can to a rimmed carving board or clean platter. Carve chicken and serve, adding more salt and pepper to taste.