Having explained this it leaves me with the comment that I can hardly imagine a situation when you not want a hat on your track. Those situations are really rare...
After our first 2p play both sheets were nearly filled and the game was longer than I would like it to be... So this is not the way how it is supposed to be played, I believe
At first I thought that if the hats were worth 2p, it might speed it up but it might be off balance?
So I guess this should be something like race for the hats, with pauses to fill other places too? :-)