Even if the meaning is to some extent clear from "How Aql is different from Qiyas?" but a clear example or two to make clear how this source of fiqh is applied would be very helpful to understand it and this is the goal of this question.
I'm honestly assuming according the only given example that it is more or less part of what sunni scholars may call maslaha or istihsaan or covered in the fiqh rules...
- The basic fiqh rule الضرورات تبيح المحضورات "the (forced) necessities allow the (application of) unlawful (things/acts)" based on (2:173) would already cover a solution for that example issue!
- Assuming this fiqh rule doesn't exist, the hanfis might solve the issue by their osol like trickery and istihssan, the malikis by istihissan and maslaha al mursala and their maqasid considerations, the hanbali also via the maslaha mursala only the shafi'i and dhahiri seem not to have a direct solution in their osol, but that doesn't mean that they wouldn't have one!