Step 1: Generously season the chickens on both sides with salt and pepper.
Step 2: Make the marinade: Place the garlic, chives, chiles, onions, thyme, and allspice in a large bowl and stir to mix. Arrange half this mixture over the bottom of a baking dish just large enough to hold the chicken. Arrange the birds on top and top with the remaining garlic-chive-onion mixture. Pour the lime juice, rum, and vegetable oil over the chicken. (Alternatively, marinate the birds in a large resealable plastic bag.) Marinate, covered, in the refrigerator for 8 to 12 hours, turning the birds a couple times so they marinate evenly.
Step 3: Set up your smoker following the manufacturer’s instructions and preheat to 275 degrees F. If using a charcoal-fired smoker, place 1 sugar cane section on the coals. Arrange the remaining sugar cane sticks on the grate in parallel rows (3 rows). You’ll smoke the chicken on top of them.
Step 4: Remove the chickens from the marinade and drain well. Arrange them on the sugar cane sticks on your smoker. Strain the marinade into a saucepan and boil for 3 minutes.
Step 5: Smoke-roast the hens until bronzed with smoke on the outside and cooked through (to an internal temperature of 165 to 170 degrees F). This will take about 2 hours. Start basting the hens with the boiled marinade after 30 minutes; continue basting every 10 minutes.