Unless the escaped pet finch is very tame and accustomed to perching on your hand or head, you will not be able to coax it back to the vicinity of the cage with lures or treats. If there are other birds in the cage their calls will attract the escaped bird, which will probably be very panicky. You can’t really risk leaving the door open, though, as the other occupants will make a break for it. Your only chance of recapturing the bird is with a net (see Netting a Bird, in the Keeping Pet Finches - Handling section of this guide).
Once a finch such as this Diamond Firetail has escaped outdoors, you'll be lucky to recapture it
Recapturing an Escaped Bird Outdoors
If the finch has escaped from an outdoor bird house or shed, the only thing that will lure him back is the noise made by the other birds. Even then, it will be a very tricky task to net him. If the finch has flown further afield, you stand no chance of catching him, and can only hope that the sound of his old home in the aviary will draw him back.