You didn't say whether your chainlink run has a top on it? That is a "must have"...otherwise, many types of animals and birds can get for an easy meal.
I would think you have an ermine/weasel. They often only kill one chicken at a time and sometimes they decapitate, but often you will only find fang marks on the neck, as they first like to drink the blood. A weasel can, indeed, can get thru a hole the size of a quarter (I've seen one do it), so you will need to patch any little openings whatsoever in your coop and also put up 1/2" hardware cloth in your run if you want to let your chickens have access to the run at night.
Free-ranging, for me, has not been a problem as far as weasels go...as they *generally* like to strike at night and our dogs do help keep them away to some extent. A good barn cat will hunt them, as well.
Another thing I've found is that chicken breeds that are quite agile and wily (such as Norwegian Jaerhons and Ameraucanas or EEs) seem to do a lot better at evading weasels.
Good luck. Weasels can be difficult to trap but I would try a "Hav-A-Hart" trap (ACE hardware carries them) with fresh meat as bait. When you catch the little bugger, take it several miles away to a wilderness area to release.
But don't wait to fortify your coop and run.