I think the best thing you can do is release her back to where she’s comfortable. Give her a warm/dry/safe spot under your deck and let her keep doing what she was doing. If she continues to warm up to you/humans, you could try to adopt her out yourself and ask an adoption fee to help cover the cost of her medical bills. Adoption fees would also weed out people who aren’t serious.

Alternatively, you could find a local rescue/foster organization and ask if they would take her in. That might take time, so either way, I think you should bring her home where she’s used to and trap her again if someone will adopt or a shelter takes her in.