Fill a martini shaker with 1 cup ice. Add 6 lime wedges, 1 1/2 tablespoon lime juice, and 2 tbsp torn fresh mint leaves. Muddle lime and mint with a pestle or a long spoon until well-bruised and fragrant.
Next combine 1/2 teaspoon warmed honey, 2 ounces rum, and 4 ounces club soda; stir gently to dissolve honey. Add to muddled mixture; stir until mixture is well-chilled. Strain into a rocks glass filled with ice; garnish with mint and lime, if desired.