Set the cream cheese out until softened, and thaw the frozen whipped topping in the refrigerator. Having both ingredients at the right temperature helps the filling come together smoothly.

Beat the softened cream cheese and granulated sugar until completely smooth and free of lumps. Scrape the sides and bottom of the bowl as needed so the mixture has an even texture.

Beat in the pumpkin puree, cinnamon, ginger, nutmeg, and salt until the color and spices are evenly distributed. Do not use pumpkin-pie filling, since it already contains sugar and spices.

Gently fold in the thawed whipped topping with a spatula until no large white streaks remain. Use a light folding motion rather than vigorous stirring to help preserve the filling's volume.

Spoon the pumpkin mixture into the prepared graham-cracker crust and smooth the top. Cover the cheesecake and refrigerate it for at least 4 hours, or overnight when possible, until firm enough to slice.

Slice the chilled cheesecake and garnish each piece with additional whipped cream, a sprinkle of cinnamon, or chopped pecans if desired. Keep any leftovers covered and refrigerated.