Stir graham crumbs, sugar, and salt in a bowl. Pour in melted butter and mix until evenly moistened.
Press the crumb mixture firmly into a 9-inch pie plate, going up the sides. Chill while you make the filling.
Beat cream cheese, powdered sugar, and vanilla until smooth and fluffy.
Whisk pudding mix with cold milk and pumpkin pie spice until thick, about 1 minute.
Stir pumpkin puree into the pudding mixture until evenly combined.
Fold the cream cheese mixture into the pumpkin base. Gently fold in most whipped topping, reserving some for garnish.
Spread filling into the chilled crust. Swirl remaining whipped topping on top. Chill until set, at least 4 hours.