- I found apocalyptically, but not with post-.
- I'm trying to say in a way that is post-apocalyptic.
Example: Post-apocalyptically-scented outdoors.
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As long as the word follows standard rules, and as long as it communicates your meaning, there is no reason why it wouldn't be valid. Remember that the dictionary doesn't define the language; it's the other way round. Also, most dictionaries will not contain every single conglomeration of prefixes (such as post-) added onto every noun, adjective, and adverb (such as apocalyptically) Your reasoning could run thus: apocalyptic is a adjective, and -ly appended to an adjective means the adverb form, so apocalyptically is a word. Post- is a prefix, and can be added to any word with which it makes sense, so post-apocalyptically is completely valid, even if it's not common.
I believe both usages work well to describe an end of the world type of smell. In a post-apocalyptic world, we can expect to smell smoke, rust, and rotten odors.
Therefore, the smell really depends on how the world ended: nuclear war, disease, invasion, zombie outbreak, etc. We can smell the world post-apocalyptically and apocalyptically. Apocalyptically is more an adjective to describe the type of smell.