That's a good question. Is the cat friendly towards people, or at least indifferent? If so, I'm confident that they will work very hard to find a good home for the cat - if they have space at the shelter. I would call them directly and carefully explain the cat's situation. If the cat is feral (not friendly or extremely fearful towards humans or aggressive), they may have other ideas of what to do with the cat - maybe trap, neuter and return - or other resources for you. Good luck!
Why don't you adopt the cat or at least foster the cat until you find it a home?
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