Connection between phenomime じ and adverb じっ‐と? (...) seems to be a phenomime, a word which evokes the idea of the state 'staring'. ‐ is an adverb meaning something along the lines of being still, quiet, focused at somebody or something, something like 'stare', in some sense. Then, it's not far fetched to suspect that these both words could be related, is one a contraction of the other, or, maybe has an adverbial been added to in some way. Is there any connection?

Yes I believe they are cognates. It wouldn't be too much to say they are the same word. This () is a suffix that follows an onomatopoeia and forms an adverb. This is often optional, but for some short words, is mandatory when the word is used adverbially.

You can find the same pattern between and , and , and , and , and so on.