Publishing expanded supplementary information as separate paper I contributed to a paper to a journal in the life sciences. The paper contains a lot of field-specific theory, but the main point relies on a result obtained through some new inference method that I worked out. Since the method was not the focus of the paper, I described it in a terse, but pretty general, supplement to the paper. The code implementation was not included.

We think this method and its implementation are of a general interest to the community, and would like to submit a separate second paper which will expand on the supplement, and provide new sample applications. In principle, there would be a lot of overlap: identical text in the derivations, similar notes, etc. Is this plagiarism? And generally, at what point would it cease being plagiarism? Obviously tacking on an introduction and a conclusion would not be okay (or would it?), but what about past that?

A related question is this: New paper which is very similar, but subtly different, from previous paper. However, the difference in my situation is that I won't be making any big additional theoretical contributions.
Plagiarism is using something without attribution, presenting it as your own. Self-plagiarism is related but a bit different. If you use your own past published work but don't cite the original it is self plagiarism but because it breaks a chain of citations that a future reader might need for context. The older work provides some context, including its own citations that are needed by future scholars. 

But you can also refer to the older work (of your own) and cite it just as you would the work of another author. In quoting, you can also be a bit more expansive, since you won't be breaking copyright rules, assuming that you still hold rights. But you need to quote, not just copy, so that the origin is clear. 

The other question is about publish-ability. That is up to the journal of course, and the new paper will be evaluated as usual based on novelty and relevance. You seem to indicate that you won't have a problem with that, so it is worth a submission.