For anyone who has experience with cats?

I have been feeding a stray cat in my neighborhood for about 2 weeks now. At first she was scared of me but she has gotten used to me since then. One day she actually started purring & rubbing up against me. But then when I called Animal Control so they could find her a home, the officer could not catch her & I think he scared her, and subsequently she is weary of me now because she saw me with him. I have still been feeding her, and she comes out & eats but then she leaves right after, and I was just wondering how I can gain her trust back? When she sees me she stays where she is but she does start to meow. When I put the food down she comes over, and she has no problem with me sitting next to her, but she hates when I try to pet her or have any physical contact. I just want her to be happy & healthy and find her a loving home but I can't do that if she doesn't trust me. I would rather not trap her because I am scared that would traumatize her even more. I tried leaving it alone for a couple days but it worries me that she is usually all by herself, huddled underneath cars all day & night. Does anyone have any advice on how I could convey to her that I am not gonna harm her? Thanks in advance :)