How do I handle a long gap on my résumé? (update)

I took time out to care for a family member after working as a warehouse supervisor. I would rather hear the honest version than the polite one. I am nervous about being filtered out. Did anyone get past it?

I listed the volunteering and course under the same dates and kept the explanation to a sentence in the cover letter.

I was hired after a 3-month gap. The manager said the skills test mattered more than the dates.