Part of the reason "ZK" has become somewhat synonomous with "SNARK" in the blockchain community is that essentially all SNARKs that any succinct proof system is already information-theoretically very close to being "ZK". Most SNARKs can be trivially modified to be zero-knowledge by something like "introduce a few redundant random operations somewhere in the circuit being proved". This is not entirely general though.

I recently wrote a blog post partially about this. What I am interested in is a generic transformation that takes a black-box SNARK and zk-ifies it without introducing any additional cryptographic assumptions. The problem might not be very well-defined but feel free to ask questions.