A dead deer found the next day should be fine if it's not over 45 degrees or so or hasn't been in direct sunlight. Especially if you shoot it in the afternoon and find it the next morning. No matter how warm it is, check the wound entry/exit for fly eggs and ticks. Now, if you couldn't find the deer, chances are that there wasn't a "good kill shot" executed, So, any gut shot animal really needs to be inspected for meat contamination. You'll notice the smell, and meat discoloration when you gut it. The affected areas can be cut out if you get to it quick enough.
That said about deer. -Basically you can jerky deer on the hoof if you can keep the flies away. Not true with Elk. Get that one at least quartered and to the deep freeze as soon as possible unless you've got cold temperatures. Bear doesn't go bad nearly as fast either.
-Steve