In a mixing bowl, combine the salt, black pepper, garlic powder, and smoked paprika; toss the drumsticks in the spice mixture until evenly coated.
Pour 1/2 cup water or chicken broth into the Instant Pot and place the trivet or steaming rack inside if using.
Arrange the seasoned drumsticks on the trivet or directly in the liquid if not using a trivet.
Close the lid and set the valve to Sealing. Cook on High Pressure for 10 minutes.
Allow a natural pressure release for 5 minutes, then perform a quick release to release any remaining pressure; carefully open the lid.
Remove the drumsticks with tongs and place them in a bowl or on a tray, then toss evenly with 1 cup BBQ sauce before serving.