Here's some advice for this situation.
If the dog's owner goes searching on the trail, that person should try to move relatively slowly and stop frequently to call and wait for several minutes. If the dog is downwind, it will take some time for the dog to reach the source of the owner's scent. If you're moving too fast, you're harder to catch up with. Be aware of the wind direction at all times and try to position the owner(s) for maximum effect in broadcasting the scent.
If your field searching comes to naught and you must leave the trail, hang clothing from the dog's owner at the trailhead. If possible, station someone there around the clock. If not possible, put food in a bowl at the clothing and check back as often as possible.
Be sure to post flyers with a photo (if possible) and a reward offer at the trailhead. Make sure you include info on the dog's name and how to contact the owner 24/7.