Around the area I live (I live in Northern Kentucky) I can think of a great amount of creatures that'd make great additions to the game as huntable/trappable food sources:
(The Obvious) Deer Deer are very populous around here and would only get moreso without more hunters. They're also very quiet and would not rouse attention from wandering stiffs. Deer meat is absolutely awesome for cooking, but there is a danger one should be wary of - in certain seasons, Lyme disease is a huge risk. If the inherent danger of contamination is taken into consideration, this could make hunting deer in Project Zomboid risky but rewarding, as that is probably going to be the only source of red meat available after power goes out. Deer meat could go toward any recipe that uses raw red meat. Potential Loot: Deer Meat, Antler, Hide, Innards [Fishing/Compost?]
Wild Turkeys Wild turkeys range all around the area I live, one can even see them crossing roads sometimes in their little flocks. Good meat, and plenty more on them than any chicken. There's also less disease risk than, say, Deer - but eating raw would still be higher chance of sickness than starter, fridge found meats. A zombie might have a hard time even getting close to a Turkey given how easy to spook they are. Potential Loot: Whole Turkey, Feathers, Innards [Fishing/Compost?]
Hungry Coyotes & Feral Dogs Kentucky, and many other places, are lousy with Coyotes since they got reintroduced after about being wiped out. Now, especially in my county, there's literally open hunt on Coyotes to keep the populations in check (and it's still not very effective). Also, tons of people keep Dogs here - they're a lot more popular than Cats, at least in my area, and there'd be a ton of ownerless dogs turning feral. Depending on how lore is determined for Project Zomboid, these wild dogs could be more hungrier or less hungrier. In the Walking Dead Comics, most predators like feral dogs or birds can still eat Zombies, so they'd be a lot less on the hungry side and breed in large numbers. If we're going by Max Brooks/NoTLD staple, animals avoid stiffs like the plague they are, so they'd be a lot more hungry and brazen (and likely dangerous to players). Either way, Coyotes and Feral Dogs could be a source of red meat that might keep a player alive - as well as a threat while travelling, especially in wooded areas. Given that they are capable of being very fast, relatively social, and might have been eating zombies (or at least, scavenging filth), bites could cause all range of diseases and death. Big survival challenge. Potential Loot: Dogmeat, Hide, Innards [Fishing/Compost?]
(Feral) Cats, Rabbits, Groundhogs, and Squirrels As unpalatable as it may be, there will be tons of feral housecats in a Zombie Apocalypse. While cats aren't as popular in my area, there are many spots I do know of where stray cats are extremely common and would be a much more accessable food source than most game (let's not get soft here, folks, this game is about survival). Consider also that cats will undoubtedly be trapped in houses when their owners are zombified and could turn quite feral without food. Scratches from feral cats could get infected easily and be a challenge to the player. I'd imagine that most cats would generally flee zombies to mixed effect, and do breed fairly rapidly. Likewise, small mammals like Rabbits, Groundhogs, and Squirrels are common even in towns out here. They'd be very spookable and flee from zombies or non-stealthing players, but with the right means (Such as traps) could provide handy foodsources. Potential Loot: Mystery Meat, Pelt, Innards [Fishing/Compost?]
(Feral) Chicken & Wild Fowl Many people near me have chickens they just let freerange on their land, and the things usually stick to yards. Zombies would be attracted to eating the chickens and could overwhelm them, but most chickens will flap/flutter away easily without anyone to keep their wings trimmed. Likewise, Ducks/Geese are common, but so are ground Pheasants and other small birds that are quite edible. It'd be nice to include nests with lootable eggs, so that players can have access to such things after power goes out. As cliche as it is, one shouldn't have a game that takes place in Kentucky without the ability to make Fried Chicken. Potential Loot: Whole Fowl, Feathers, Innards [Fishing/Compost?]
Pondlife & Fish A fishing system should definitely be included in Project Zomboid, maybe even with its own skill. Fishing is very common around Kentucky. There's plenty to catch, from Bluegill, Catfish, Bass, to Gars (*which aren't very useful for meat, but hey, they're there to annoy), even large Crawdads. Heck, there are bullfrogs in my ponds that weigh as much as cornish hens with legs like small drumsticks. I'd absolutely love for these kinds of things to be catchable in Project Zomboid for a character to survive off of. Zombies would not have access to most of these creatures and thus could provide a relatively safe foodsource for players willing to invest the time/luck. Potential Loot: Fish, Fish Bones, Innards [Fishing/Compost?], Dead Frog, Crawdad
I'm hesitant to include Wild Boar and Elk on here, as they don't to my knowledge tend to get close to suburban/urban centers. I do know that Wild Boar are getting brazen and would only get moreso during a Zombie apocalypse, though - they might be an additional consideration.