You want the points of the jester hat to respond to the head bounce?
A picture is worth a thousand words, especially since I’m a slow typist and easily confused by words.
Of the top of my head, I’m thinking an expression on CC Bender, or if you have CS3 maybe an expression controlling a Puppet Pin (correct terminology??) of the Puppet Tool. But after a quick test, and I think Dan’s Squash and Stretch looks pretty good. Move the anchor point of the hat layer to the bottom of the hat.
https://www.motionscript.com/expressions-lab-ae65/squash-and-stretch.html
maxDev = 33; // max deviation in pixels
spd = 30; //speed of oscillation
decay = 1.0; //how fast it slows down
t = time – inPoint;
x = scale[0] + maxDev*Math.sin(spd*t)/Math.exp(decay*t);
y = scale[0]*scale[1]/x;
[x,y]
Keep the “spd “of Squash and Stretch the same as the “veloc” of Jack In the Box, and keep the decay the same for both.
In fact Squash and Stretch might be nice on jack too, and if the hat is parented to the Jack body, no Squash and Stretch would be needed on the hat, just an option.