Two things, a good bit and clear cues.
Honestly I've found most horses figure out how to neck rein rather quickly with a short shanked, ported or mullened, curb bit. I use a bumper sweetwater bit:
If you want to stick with a snaffle then essentially you're teaching the horse a pre-cue, the rein touches the horse's neck, if they don't turn you direct rein them. Do this enough and eventually they'll turn with just the pre-cue, before you need to pull. But that's essentially what a curb bit does on it's own.
What happens with a curb bit while a horse is learning - you neck rein left, your right rein is shortened, this pulls the right shank up and takes the pressure off the right bar - pushing onto the right cheek, which pushes pressure down on the left bar. This provides a number of very clear signals for a horse to follow when you neck rein, the horse will also learn the pre-cue of the rein touching the neck before the contact on the bit happens.
So if the horse gives well to direct reining it could be as simple as changing to a good quality, gentle, curb bit. But if you want to stick with the snaffle your best bet is to train the pre-cue.
Good luck - have fun!
ETA: Also - welcome to the forum!! we love pony pics here so feel free to post up