Could you please elaborate on the jump from "all" to "some" here?

It's very hard to view "being hated by all men means that some man will do something consistent with hate" as anything but a blatant corruption of the meaning of "all".

Note also that in SC's case, upon querying, that she believes the perception is accurate--otherwise you could say, "Well, if any man (you can't predict which in advance) might do something bad, it feels the same operationally as if every may would do something bad". So you can't make the leap from "some" to "all" by leaning on the feeling part of it. (For people who are not SC, this may be a valid move, however.)

Also, while imprecise language might be excusable inasmuch as expressing oneself is difficult at the best of times and moreso when a topic may be emotionally fraught, I don't think we should encourage it.

The stance that you describe would more reasonably be described as "feeling in danger of hateful actions from arbitrary men" or somesuch.

The reason this is important is that if one is going to follow on from this observation by using any reasoning, what one does with "all" vs. "arbitrary" is markedly different. If there is such a thing as character, and it can be discerned to a sufficient extent, and character substantially impacts disposition, then in case of "arbitrary" selecting the right men to be around is of utmost importance, as is fostering the development of men of suitable character. In contrast, if it actually is "all", then neither selection nor training is likely to be fruitful and instead one should take a structurally defensive approach to avoid harm despite constant threat.

(Would this be clearer if I explicitly reference the distinction between universal and existential claims?)

This has pervasive and profound consequences for how one would live one's life, and how we should structure society. So I don't think it's fair to pass off language that says one but with only support for the latter.