Agree with others, mostly Sorrel & CowChick. *Assuming* he's comfortable in the bit he has now, no need to change. If it's too big for him or such, changing to a french link snaffle would be my usual choice of bit. I don't believe adding further pain in his mouth is generally helpful. Remember he is 20yo & may have had a lifetime of 'practicing' this behaviour & resisting bit pressure, so using a bitless can often be more effective.
First & foremost though, before treating it simply as a training problem, I'd want to rule out/treat any pain issues, as this sort of behaviour is often 'running away from pain'. Eg. Saddle fit may be problematic at a canter, etc. May also be riding style/balance/ability at fault, rather than the horse just not having been trained well.
People generally have the idea that a normal single link snaffle is not a harsh bit. While I do agree that it is as much about the hands on the reins as anything, I beg to differ, that a regular snaffle, especially if used with equal rein pressure on both sides, can be quite painful, with it's 'nutcracker' effect.
Re the 'run them until they want to stop then ask for more', I agree that this can be good... in theory. However I've known horses that will run all day if you allow them to & it also depends on your environment/ability. I think this can be good for some things, but not for this sort of behaviour - it also doesn't teach them to stop when you want to, only to listen to you when they're ready already IMO.