That's how we got our first cat. Found her outside and went door to door looking to see if she belonged to any of the neighbors.
We've had lots of cats since then. Unfortunately, the first one died several years ago. We think she was poisoned. We've had 3 others disappear. We have a lot of coyotes around us, and sometimes they've just gotten out when a door was opened.
We have 3 now and they've been with us between 7 and 10 years. They really don't leave the yard (which is fenced, but coyotes will find a way). It's hard...we don't like letting them out, but for an hour or so each day, they all want to go out and lie in the yard.
Best of luck finding a home for her. She looks like she's a sweetie! Perhaps she can be an indoor kitty for you and you can just keep her!!