Assume he is nearer and quieter than you expect
A green-cheeked conure does not announce himself like a big parrot. He is a small, mostly green bird with a maroon tail and a call that carries only a short distance, and a frightened one often freezes in foliage rather than flying far. For the first hour, work on the assumption that he is within your garden or the next few gardens, tucked into a tree, hedge, or climbing plant where his green back disappears against the leaves.
Walk slowly and scan in layers: low shrubs first, then mid-height branches, then the canopy. Look for the wrong shape rather than the right colour — a small round silhouette, a long tail hanging below a branch, a slight movement when everything else is still. Binoculars help enormously here, even at short range, because a conure sitting motionless can be invisible to a hurried glance two metres away.
Call, then give silence a chance to answer
Call his name, his usual whistle, and any kiss or click sounds he responds to during play at home, then stand still and listen. A green-cheek may answer with a soft, short call you would miss while walking or talking. Repeat the pattern — call, pause, listen, move a few steps — around the whole perimeter of the last-seen spot before you expand outward.
If he has a bonded cage mate or another bird he chatters with, bring that bird out in a secure cage and let the familiar chatter do some calling for you. Never carry a bird outside unsecured as a lure, however well behaved he seems; one startled flutter would leave you searching for two birds instead of one.
Set his cage where a small bird can drop to it
Place his open cage in clear view of the trees you are searching, raised on a table if the ground is cluttered, with favourite treats and his usual food arranged at the open door. Green-cheeks are nimble, acrobatic fliers, so a bird who spots home below can usually make the descent — what he needs is to see it. Stay close enough to close the door calmly when he goes in, but do not hover over the cage or lunge the moment he lands on it.
Check the ground layer too. A young, clipped, or exhausted conure may end up low in grass or under a bush rather than up high, so part shrubs gently by hand as you pass and look under garden furniture, woodpiles, and decks.
Get more eyes onto the greenery quickly
Knock on neighbours' doors within the hour and ask them to check their trees and to listen rather than chase; give them a photo that shows his green body, grey-scaled chest, and maroon tail, since 'small green parrot' is easy to confuse with other species. Call local vets, shelters, and bird rescues with the same description, and post to neighbourhood groups asking for sightings with a time and a tree, not capture attempts by strangers.
Keep a simple list of who has searched which garden and any calls heard, so effort is not wasted rechecking the same hedge while an unchecked street goes quiet, and use KinsLe's Lost Pet Alert System to push a free alert to nearby members who can message you directly with a sighting.