My coop is one of those "prefab" for 12 chickens or so...it's really not much but sleeping quarters and laying areas - so putting food in there isn't really an alternative
You could use a chicken feeder (or a bird feeder!) outside the coop. It could keep the feed off the ground, and you'd have the option to bring it indoors at night so nothing else eats the feed. It could hang from a tree or from a post, sit on a stand, attach to a fence, or perhaps sit directly on the ground--depends on the style of feeder, and on what is available. Hanging feeders often work really well, because you can adjust the height so the chickens cannot scratch feed out with their feet (but can still reach easily to eat.)
It sounds like a lot of the feed is getting wasted (not eaten by the chickens), so a feeder might also save you money on feed.