Caring for an outdoor cat with no home?

i am going through exactly the same thing right now , so i feel what you are feeling .i have a beautiful little girl that has  been coming to my house  since late may . she comes off and on throughout the day but i rarely see her after 7 or 8 pm , but the very next morning there she is waiting on my lawn chair ( i put out a comfy blanket for her to lie on )  for her food . she is so clean and calm but i am going through the same thing as you . she never runs when i open my door , she takes her treats directly out of hands but when i go to touch her she runs away , but not off my patio just away from my youch . it's killing me because she has become very important to me . i have posted pics of her all over facebook . no one is claiming her and no one says they recognize her . i am caught between a rock and hard place . because if i take her in and keep her she may belong to someone and if don't take her in she may be outside in the freezing cold all winter . ( and that would tear me apart )  what i'm wondering is , where do she go every night ? does she have a home or does she have a " safe place " to huddle down for the night . i'm wondering where she is so comfy all day on her chair on her furry blanket wouldn't she just stay there at night as well is she  was homeless ?

The chair may feel comfortable to her in the day but she may go somewhere where she feels safer at night.   Can you set something up on the porch - a small doghouse or cat shelter?  It would give her more protection.  That won't work well if there is another cat (or other critter) around at night.  He or she may be spooking her away from your porch.  Make sure you don't leave any food out after dark.

Also, during the winter, the blanket will wick up moisture and become useless.  If you fill the shelter with straw, it will give her a comfy place to bed down.

If there are other critters spooking her at night and if no one has claimed her, is there a possibility of you trapping her and getting her vetted, then allowing her to live inside your house?  No matter what, that is the safest place for her.