SET THE CAGE THEN LEAVE IT,

you have to get to the third island first, that literally was the only time I found my
cage had caught a chicken after I reached the third island, but I still had the cage on
the second island >.>, and I talked to a friend she didn't catch hers either until
the third island.

If that doesn't work make more cages, and set them in more places.

If you don't know where to lay this trap then its where ever you picked up chicken
feathers from in the first place.