Check out Greyhound Downs 101 and Greyhound Downs 102 if you have not already.
I want to remind you that once your greyhound is down, continue to reward him for maintaining the down. Do not ask him to stay yet, but simply reward it for as long as he offers. I like to mark these moments with a clicker or a "yes". Both mean "that is right and I am giving you a treat for it".
At some point your greyhound may realize that lying there is a pretty good deal and resolves to stay there forever. Time to teach a release word. I use "done". "Done" means you are free to go.
When I teach a release word, I touch the collar with one hand, lure the dog out of the down with a treat in my other hand, and say "done".
Get in the habit of releasing your greyhound, so he does not develop a habit of releasing himself. So if you think your greyhound is about to pop up, release him. You are finished training or ran out of treats, release him.