Here is a sentence from an English magazine:
With the weather still on this side of chilly, it might be better to stay in and nest ― especially when nursing a cold.
Why not write on this side of chill? Can we always add an adjective after a preposition?
The sentence from the magazine is informal, even a bit ambiguous, but it is correct insofar as it uses chilly rather than chill. One might well say that The weather is chilly or It's chilly outside. One might also say There is a chill in the air, but one would never say The weather is chill.
Therefore the word chilly is the correct one for describing weather in your magazine's sentence.
A preposition is usually followed by a noun phrase. 'this side of chilly' is a noun phrase.
