Step one, of course, is catching the cat. This will be a million times easier if you get your hands on a live catch trap. You can buy them, online and in stores, rent them, or make your own. Google, "live catch trap", and see what options you have. If you want to catch this cat in the safest, least traumatizing way possible, you need a live catch trap. I got one from Lowe's for $50, and it was worth every penny. It is an investment, especially if there are multiple cats you want to catch, especially if you plan on catching more in the future. They are sturdy, they can be reused over and over again, and they are safe for the cats and safe for you.
Step two is socializing the cat.
It is a myth that adult feral cats cannot be socialized.
It is a myth adult feral cats cannot be happy indoors.
Please trust me when I tell you that as long as you are patient and kind and treat them with all the compassion they deserve, you can help them.
These are traumatized animals, they won't get over their trauma within a day of being indoors. You have to consistently treat them with compassion, kindness, and respect.
But you can help them. They can and they will be safe and happy indoors as long as you give them the chance to realize that they are safe now.
I cannot stress it enough that these animals are traumatized, what they need is help, not to be released back into their traumatic environment.
If you are able to keep the feral cats that you catch, please do. Treat them with compassion and kindness and they will learn that you are not a predator, they will learn that you are more than a source of food, they will learn that it is safe to love you.
If you can't keep them, please bring them to a shelter, or find someone that is willing to take them in. Please do not release them back into the wild.
Trap-neuter-return is the equivalent of rescuing a drowning person from the water, giving them CPR, drying them off, and then shoving them right back in even though they still can't swim and they're exhausted.
A cat will always be safer and happier indoors, and they will also no longer be harming the native species in your area. Cats have been a domesticated species for thousands of years, there is nowhere on this planet that they are not invasive. The native animals in your area that these cats are killing to feed themselves have no defense against them, and cats have already driven several species to Extinction.
Rescued ferals will likely not be exactly as cuddly as a normal non traumatized cat that has been raised and handled by humans since birth. ( I mean, there's every chance they could be – it depends on their personality)
But they will love you, they will be happy, and even if they don't sit on your lap, they will follow you from room to room just because they want to see what you're doing. They will purr when they notice that you are looking at them, they will come to the door to greet you when you get home.
If you are able to keep the feral cats that you catch, please do. Please know that it is a myth that they cannot be happy indoors, that it is a myth that they can never be tamed.
They are traumatized animals, and if you show them kindness and compassion and respect and give them an environment that is safe and free of dangers, they will heal.
It might take say, three years. But they will heal. They will be happy. They will be safe. They will learn that it is safe for them to trust you, they will learn that you are not a predator, they will learn that you are not cruel, they will learn that it is safe to love you.
It is a myth that adult feral cats cannot be socialized. It just takes more time, because they are more traumatized then their kittens. It is possible, as long as you are patient, as long as you respect the animal, and show them kindness.
Some tips for socializing adult feral cats
Take off your shoes. I am 100% serious. Especially if you wear heavy boots. Take off your shoes. You, from a little tiny cats perspective are a giant. When you are running around the house in giant heavy boots, you are a giant who is incredibly loud and threatening. Put yourself in the cat position. Would you be more afraid of a giant dozens of times taller than you if they were quiet and walked softly, or if they stomped around without a care in the world? Take off your shoes, make sure that when you are going up stairs you are not sounding like a herd of elephants.
When you walk near the cat, make sure that you are giving it plenty of space. Again, you are a giant, it is afraid that you are going to step on it. You need to make it clear with your body language, with your verbal language, with everything possible that you are not going to step on it, that you are giving it plenty of space, that you are aware that it is there, and that you are trying to be respectful. Animals can understand body language just like we do. If you make it clear that you are being careful of the cat, the cat will know this, and it will be less frightened of you when you walk near it.
If you have little kids, do not let them near the cat until you have explained the proper way to treat a cat, and explained this cat is very frightened and needs reassurance and that you cannot grab the cat or try to pick it up.
Do not attempt to pick up the cat ever unless it is a literal emergency. It's one thing for kittens who were hand raised, it's another thing entirely for a traumatized animal to suddenly be grabbed around the middle by a giant and picked up through the air. A cat's stomach is one of its most vulnerable areas. Even hand raised kittens will feel anxiety when they are touched on their belly without warning or if they aren't used to it. Only real comparison I can come up with here is if a random stranger walked up and like, started tickling you or something. Except worse than that, because as far as this cat is aware, you are a predator that wants to eat it.
Do not chase the cat around the house and corner it and force it to sit still while you pet it. Your cat has bodily autonomy just like everybody else, and it is not okay to violate that just because you want to pet it. You have to treat this animal with the same respect you would give any other animal, or any other human.
This sounds dumb, but if you have to vacuum? Warn your cat ahead of time. Say, I'm I'm turning the vacuum on! Or something like that, a few minutes ahead of time every time you plan to vacuum. Use the same words, the same intonation. not only will this keep them from being startled, they will learn that the the scary loud noise isn't random, it can be prepared for, and if you're not literally chasing your cat around the house with the vacuum, they will get used to it! When you turn the vacuum off and are sure that you don't need to turn it back on, say so! Say all done, or if the vacuums off, every time. Same words, same intonation. Train them to its meaning. As long as you warn them ahead of time, and consistently reassure them while the vacuum is on, and make sure they know when it is over? they will become less afraid of the vacuum over time and eventually they just won't care anymore. This also works with dogs! As long as they are aware that it is going to happen, as long as you don't ever traumatize them with the vacuum, they won't be afraid. Or at least, when they're done hiding from the loud scary noise, they will come out that much sooner. They will know the scariness is over. Animals are smarter than we give them credit for. If you warn them, make sure they know it's safe, and make sure they know when it is over, they won't be as afraid as they would if this random scary sound just happened out of nowhere.
Have a specific phrase you use when you are opening a can of wet food for them. This serves two purposes. One, they will come to you when you say this phrase and hold up a can for them. Two, it might stop them from swarming into the kitchen every time you open a can of soup.
But seriously. Treat them like you would treat a person. Be respectful of their fear, they fear for a reason. Be respectful of their personal space, be respectful of their boundaries, be kind, be compassionate, make them feel safe.
Be patient. If you care about the cat, if you want what is best for the cat, if you want them to live a long healthy happy life? Be patient, give them all the love you have, and they will be better for it.