In a medium bowl, combine pumpkin, ½ container whipped topping, ½ cup confectioners’ sugar, pumpkin pie spice, cinnamon, and salt. Beat at medium speed with a mixer until well combined; set aside.
In a medium bowl, combine cream cheese, remaining ½ container whipped topping, remaining ½ cup confectioners’ sugar, and vanilla extract. Beat at medium speed with a mixer until smooth.
In small glasses, layer cream cheese mixture, gingersnap cookie crumbs, and pumpkin mixture. Garnish with whipped cream, pumpkin pie spice, and gingersnaps, if desired.