A single jointed snaffle has a nutcracker effect on the tongue, as well as possibly gouging the roof of the mouth when both reins are pulled with much force. Depending on the level and type of riding, it can be far from the 'gentlest' or most appropriate bit. That said...
It sounds like you may have a training issue rather than, or as well as the bit type. It's possible that the horse hasn't ever been properly desensitised to a bit, if he plays & fidgets with them regardless what's happening. Without knowing many details, I can only guess, but out on the trail perhaps he's eager or excitable, or wants to stop & graze & he's 'rooting' at the reins? Perhaps you're also holding the reins too firmly or constantly in anticipation & he's fighting against that? Whatever, I think this is about your effective communication with eachother, not so much what equipment you choose to help you.
I would suggest working on desensitising him to a bit - the mildest. Put the bridle on him & just let him wear it(supervised) without being hassled with pressure. Use a halter & lead if you want to do stuff with him. Don't ever take the bridle off when he's fidgeting if you can help it, but wait until he's quiet a moment or more.
*After* he's truely comfortable with it, then I'd start asking for basic stuff with the reins. Its a balancing act between reinforcing the right behaviour & not reinforcing the wrong, but as much as possible you need to also strive not to release/reinforce when he's fidgeting. I would teach him this rather than using a noseband to tie his mouth shut, effectively saying 'put up & shut up'. This way, once he's desensitised to wearing it, there's no stress for him, he can relax. Plus if later he starts to play or gape or resist again, you'll be able to 'hear' him, knowing there must be a problem, such as pain, stress, the rider, whatever.
I would be playing with him in a halter and lead in the meantime, mainly on the ground at first, to get some effective communication and leadership(you, preferably
) happening. It will help your whole relationship, but you will also teach him to yield softly, teach or reinforce yourself for being effective, considerate, having good timing, reading him, so you will both get in tune with each other better when it comes to bits.