Pop tab off beer can. Pour half of beer (3/4 cup) over soaking wood chips or chunks, or reserve for another use. Use church-key-style opener to poke 2 more holes in top of can. Set aside.
Remove packet of giblets from body cavity of chicken. Remove and discard fat just inside body and neck cavities. Rinse chicken in cold water; drain and blot dry, inside and out, with paper towels.

Sprinkle 1 teaspoon of rub inside body cavity and 1/2 teaspoon inside neck cavity. Drizzle oil over outside of bird and rub or brush all over skin. Sprinkle with 1 tablespoon rub and rub it all over skin. Spoon remaining 1 1/2 teaspoons of rub into beer through hole in top of can. Don't worry if beer foams up; this is normal.
Hold bird upright, with opening of body cavity at bottom, and lower it onto beer can so the can fits into the cavity. Pull chicken legs forward to form a sort of tripod, so bird stands upright.
Tuck tips of wings behind chicken's back.

Set up grill for indirect grilling (if using charcoal grill, dump or rake glowing coals in 2 piles at opposite sides of grill and place drip pan in middle; if using 2-burner gas grill, light 1 side and put chicken on unlighted side). If using gas grill, place wood chips or chunks in smoker bowl or in smoke pouch and heat on high until you see smoke, then reduce heat to medium.
When ready to cook, if using charcoal grill, toss all wood chips on coals.

Stand chicken up in center of hot grate over drip pan and away from heat (if using charcoal grill).
Cover grill and cook chicken until skin is dark golden brown and very crisp, and meat is cooked through (about 180 degrees on instant-read thermometer inserted in thickest part of thigh, but not touching bone), 1 1/4 to 1 1/2 hours. If using charcoal grill, you'll need to add 12 fresh coals per side after 1 hour. If chicken skin starts to brown too much, loosely tent bird with aluminum foil.

Using long tongs, hold chicken by can and, using another set of tongs (to steady chicken), carefully transfer chicken and can in upright position to platter. Let chicken rest 5 minutes.
Carefully lift chicken off can: Work over sink or roasting pan to catch any liquid. Wearing heatproof gloves, hold chicken slightly at angle with 1 hand, carefully pull out can with other hand (you may need to twist or wiggle it to loosen it). To prevent burns, take care not to spill hot beer. Or, you can hold chicken with tongs and remove can with another set of tongs.
Halve, quarter or carve chicken and serve.

*Cook's note: For a variation, use Iced Tea Rub and cook chicken on 12-ounce can of iced tea. To make Iced Tea Rub, combine 1 tablespoon powdered iced tea mix, 1 tablespoon paprika, 2 teaspoons ground coriander, 1 teaspoon sugar, 1 teaspoon coarse salt (kosher or sea salt), 1/2 teaspoon freshly ground black pepper, 1/2 teaspoon garlic powder, and 1/4 teaspoon celery seed. Stir to combine. Follow directions for Basic Beer Can Chicken. Store remainder in airtight jar.

Put salt, brown sugar, paprika and pepper in small bowl and stir to mix. Your fingers work better for mixing rub than spoon or whisk does.
Store rub in airtight jar away from heat and light; it will keep for at least 6 months.