I have somehow managed to overlook this thread.
A couple of years ago I reunited a cat, Zsa Zsa, with her family after she had been missing from home and living rough for three months. She just turned up in our garden one day: she'd been living in a little open area nearby, apparently. She was in perfect health, just a little thinner than when she'd sauntered off. Her family had given up hope - try not to do that
One thing - this little open area was only about 500 yards from Zsa Zsa's home, in a more-or-less straight line, but it was the other side of a churchyard and a warren of back gardens. So I would say, try to imagine Lilly's possible routes from a cat's point of view, rather than a human one.