I doubt the mini produces enough heat radiating through the bottom of the unit to make any noticable effect with that phase changing pad. It requires hours of use with a 15" laptop to produce the effect it's marketing, now extrapolate that to the power output of the mini vs a common 15" notebook.
Obviously the air flow in the unit is not optimal for convection, or the bottom of the unit wouldn't get so hot. The bottom vents should have a net air flow that is sucking air into it. If you want to have any noticeable effect on cooling the hardware, you either have to cool the air down (from ambient) or increase the velocity of the air moving through the unit.
If you want to stay silent, you really only have two options.
1. add or expand vents throughout the chassis. If you're willing to do this, then you might as well upgrade the heat spreader across your chips and add any heatsinks you can to any other chip on the board that you can fit.
2. use peltiers via an external pad sitting just below the bottom vents. This will allow you to use a large passive heatsink on the hot side exposed directly to outside air and away from any intake vents of the mini, while cooling the air right at the bottom vents significantly. The temp difference between hot and cold inside and directly outside of the vents will have two effects, 1, the density difference will increase the air velocity into the mini, and 2, the temperature difference will obviously remove more heat from the hardware. mimicking running your mini in an air conditioned room vs ambient.
Also, if you haven't noticed your mini getting hot then you either haven't been using it or you always use it in AC. in 80-90F ambient temps i've had it up in the 74C range while sitting on a flat surface, or while just hovered in air in my hand.