Find a Paw
People tend to search for a missing cat the way they would search for a missing dog. The two animals behave almost nothing alike when frightened, and the difference changes where you should be spending your first afternoon.
Cats usually stay much closer than you expect
An indoor cat that gets outside is generally not exploring. It is hiding, often within a few gardens of home, and often in total silence. Cats in this state frequently do not respond to their name, which leads owners to conclude the cat has gone far away when it is in fact under a neighbour’s decking.
So the search is a slow, close, physical one. Sheds, garages, under cars, behind bins, beneath decking, inside hedges, up trees. Take a torch even in daylight, because you are looking into dark gaps. Go out last thing at night and first thing in the morning when it is quiet, and stop frequently to listen rather than call.
The single most useful thing you can do is ask every immediate neighbour to open their shed and garage and physically look. Cats get shut in constantly.
Dogs generally travel
A dog that bolts, particularly one startled by fireworks or traffic, can cover ground fast and in a straight line. The relevant search area grows by the hour, and it grows outward along paths of least resistance: roads, tracks, footpaths, the edges of fields.
This is why telling people matters more than walking for a dog. You cannot cover the area on foot, but a few hundred people in the right postcode can. Contact vets and shelters early, because a loose dog is far more likely than a cat to be picked up by a stranger and handed in.
If you get a confirmed sighting, go to that spot and stay there with something that smells of home. Chasing rarely works. Sitting still, low to the ground, not making eye contact and letting the dog come to you often does.
Both: think about where they were going
Pets that go missing from a new home very often head towards the old one. Pets with a favourite walk head for it. Pets that are food-motivated head for wherever food has ever come from.
It is worth five minutes with a map, thinking about what your specific animal wants, rather than searching outward in circles.
How wide should you tell people?
Wider than you have searched. The whole point of alerting an area is that it covers ground you physically cannot, at the same time as you are covering the ground you can.