Whisk together the pudding mix, espresso powder, pumpkin pie spice and salt. Add the milk and pumpkin purée. Beat with the whisk until thickened, about 2 minutes, then let rest for 5 minutes.
While the pudding rests, crumble the ginger snaps. Add about 1 tablespoon of the crumbled cookies into each of 4 coffee cups or glasses. Add 2 tablespoons of pudding on top of the cookies in each cup. Then add another layer of cookies and another layer of pudding. Top each cup with a dollop of whipped cream and sprinkle a few more cookie crumbs on top. Serve immediately or refrigerate for up to 4 hours.