The trick is to minimize the contact area between the just varnished piece, and whatever it's resting on.
Having the piece rest on an edge minimizes the area. And the best way to get an edge is to just take a piece of hardwood, about 1x1 cm or larger, and slit it diagonally through, resulting in two triangular cross-sections.
These will still stick to the wood a little bit, and sometimes leave a visible
mark in the varnish, though one that will not be visible after the next coat.
I always let it dry with the "good" more visible side up anyway, so this doesn't matter that much.
Here's a closer view. Incidentally, the funny piece of wood I varnished was a "planer holder", to go on my wall with all my other tool holders. The gears are part of my box joint jig.