Never give up.
Posters with contact details (online at lost and found pet sites and services/on the street level/in the local bulletin boards in shop windows or big supermarkets).
Do the leg work.
Work with what you know about the cat.
If it's a new cat you've just adopted and has ran away (so the cat would have no reason to return to you since it has nothing to connect you or it's new home as home), always inform any previous owner if you've a 'displaced lost cat', never be afraid to say
"You know that cat you've just sold me?.."
"It got out my house right away/I under estimated it and the window..."
"And now I can't find it."
"Your fluffy baby I've just adopted is now lost." (Basically).
Because chances are, they'd have the cat microchipped.
If they've microchipped the cat but haven't updated the information or don't have the microchip number to hand, then which ever vet who inserted the microchip would have the microchip number on their records of when they inserted the chip - they'd always ask for the pet to be registered upon inserting the microchip and they'd always keep a record of the microchip's number inserted into the animal on its file.
It's on you to try your hardest and inform all parties with a vested interest in any "new" lost cat.
There is a whole plethora of different microchip companies with different databases and chances are; if you read the number to a microchip company, if it's not on their database, based on the number, there is a real chance they could point you in the direction of the correct microchip company. Once you've made contact with the correct company, notify them the cat is missing so they can flag it. Update the information (which should take 28 days).
Have the microchip number to hand at all times so you can use it to identify your cat if it's found within the transfer window, all the registered keepers details will be locked during this transfer, but the number will still show when scanned.
A previous owner would probably have more photos and videos of the cat you could use; try using a photo that shows the cat's body and length of fur and any unique markings as well, rather than it sitting on a pillow looking at the camera (unless it's got a unique face). Chances are, when looking for the cat, you're going to see a lot of cats, so try and find a picture where you can match body size and fur length better, if you can.
Talking to the previous owner about behavior could indicate if you've a lost timid cat or playful cat (there's a bunch of cat personality types you'd see when diving into the world of pet detective to find your missing cat).
Know what type of cat you're looking for.
The personality of the cat can let you know what radius to cover when searching for your cat.
If the cat's an indoor cat or an outdoor access cat, can also determine which radius.
After 10 days, any cat will be so in need of TLC, it won't be afraid to approach a friendly stranger, between 7 - 10 days.
Days 1 - 3, the cat is lying low, hiding, it's a basic survival strategy for the cat to keep quiet and probably isn't too far away - but it is an expert hider, master of not being found if it doesn't want to be.
Days 3 - 5 a displaced adult outdoor access cat will then feel brave enough venture out further.
Days 7 - 10, it will be so in need of TLC and want of cat food it will open up to the kindness of strangers.
To be kind to a cat.
Food. Food. Food.
This is an animal still.
It associates food with good and your smell with the food smell creates a positive association with yourself for the cat.
How to get your scent into its food?
You could handle it with your bare hands, might sound nasty or gross, but, our hands have pours and we have a smell and if we wash our clean hands in its food a little, any animal who finds this food will smell you and the food together and will probably let you hand feed it if you're lucky enough to find it once a trust is built up between you and the animal. Use plastic bowls as plastic bowls keep the scent better than ceramic when creating this positive association. Handle the plastic bowl as well as the food to get your scent all over it.
Cats are all about the positive association, so any new lost displaced cat might find your food (or another cat/wild animal), and learn your smell as good when mixed with food. So to be nice to a new lost displaced cat, the answer is food, and to create the positive association to yourself with this new displaced cat, get your smell in its food to tie you in with
'oh, this human isn't so bad, they give me food'.
And that's how to win a cat's trust.
Then depending on the personality of an outdoor access cat (a previous owner could let you know what type of cat it is). how far.
Let's say it's what's classified as 'brave cheeky cat'... Give it a mile.
Also, if the cat is spayed or neutered can also factor in its radius.
A cat will establish itself somewhere.
Probably a garden or a house of another cat/where there is a source of food and shelter.
So do the leg work, don't be afraid to knock on doors in your search area if you suspect your missing cat is around, because it's more text book than unique. Just got to work out what type of missing cat you have, do the leg work, get the poster made and circulated. Local vets and animal shelters could also do with a copy of a missing poster in case the cat is found and taken to be scanned locally.
The Cat's Protection League could let you borrow a humane cat trap as long as you return it within the agreed time.
Good luck pet detectives, because, like it or not, you're going to enter the world of pet detection when looking for your cat.
I speak from personal experience and a success story that wouldn't have happened if I hadn't have made posters and in looking for my cat in 2019, I found an identical missing cat which was reunited too with that cat trap...
Took the cat I thought was mine to be scanned, was another cat who was missing too. The microchip knows.
Get the information, update it, and let the company know the cat is missing.
My cat was found within 3 weeks by somebody else alive and well who saw a poster of mine online.
They contacted me, I got the call, to confirm we went to the vets to get it scanned (out of hours vet one Saturday night), it was mine.
I was looking in the correct radius, but in a completely other direction (you live, you learn), but the radius and personality was text book based on the personality and type of missing cat. It was an American website with very few reports in the London area that taught me about types of missing cats and classifications of missing cats and different behavioral types. I'm not sure if I can link it due to the forum, but, to pin point a search area, find some American company called Pet FBI (something to that affect) and you can register it missing there too for free. I live in Greater London, there was a handful of missing or found cats in my area on this American site, but it taught me how to determine the search area.
Once found, inform everyone with a vested interest in this cat that it's found. This may include former owners you've notified, local places you've displayed posters, the microchip company, any neighbours looking for the cat or keeping an eye out for your cat, inform everyone the cat is found once found.
There is a very high probability of a missing cat being found, so bare this in mind and do the leg work.
At dusk and dawn are great times to see cats (because that's when they like to be out I gathered from my patrols, so I began to patrol around these times), also, bring a bright torch, even in the daytime a hiding cat could be hard to see, so bring a bright torch and use your ears too, when patrolling/looking for your cat. Listen for jumping sounds and running sounds and noises the animal makes to determine what something was and where it ran off too as well as your eyes.
If you think you've found your cat. Loiter there, this gets your smell in the air.
Try to hang out around it, and bring food and a crate for it and try and lure it into a crate (or humane cat trap) with the food. If it doesn't work and you can't catch it, bring food and feed it. This lets the cat know you're friendly, and, it lets you know your cat has eaten. Try and bring fresh water too.