I'm almost positive I know what your problem is. Lynx. We lost three chickens the very same way you lost yours, except I watched it and couldn't get to them in time.
Lynxes are like bobcats, but a little more stealthy. They are somewhat nomadic creatures, but stop when they locate a good place for food. Then they remain there until the prey has run out. Lynxes attack during the day, around the time you offered, however their favorite time is before sunset. (If you're wondering, we looked this up shortly after the ambush on our chickens.
) They are extremely skilled climbers and they are very smart. For us, they climbed a tree and jumped over the fence and into our coop, then dragged our chicken over the fence. We won the fight though and were rewarded with being able to bury our lost hen. There's not much you can do against a lynx because they are a protected endangered species.
(But I am not a fan of killing anyway, I mean it's the circle of life, people.) However, I suggest you build an entire enclosure for your chickens' area and add a little more support to the fences. I don't have the best solution, but I'm pretty sure I know what the problem is. Take some binoculars and watch for a lynx. They look exactly like a cross between a small house cat and a bobcat. Good luck!
