Hard objects collide inelastically which transfers more force in the opposite direction after the collision. Adding a rag, towel, or pillow softens the surface and the collision is now elastic as some of the force of the jar colliding with the countertop is absorbed by the fibers of the cloth. Think what would happen if you dropped a glass marble on your countertop versus on a towel on your countertop. The marble would likely bounce up in the first scenario and not in the second. Similarly, the sauce in the jar experiences upward force from the jar after the inelastic collision with the countertop but not with the rag softening the elastic collision. You could optimize sauce flow to the bottom by changing the method used to apply forces to the jar. Get a string, tie it around the lid of the jar, and swing the jar around your head a few times. If you don’t yeet the jar against the wall accidentally, almost all of the paste/gel/viscous liquid should be at the bottom of the jar afterward.

Probably works in the same way as a dead-blow hammer, i.e. dampening rebound from the hard surface.

I'm not a physicist, but I'd imagine the difference is between a sharp slow down and a more gradual one. When you're using the wet rag, the point of impact is a few milliseconds longer than when rapping it on a hard surface.

I think it's because a sharp blow will cause a single large wave front to propagate through the medium.

And with a dull blow, probably multiple smaller wave fronts that can interfere and create mini pressure differentials thus more settling

What you are describing is a non Newtonian fluid. Maybe your paste has additives?