He kind of sounds like he isn't very tired. You let him scream between midnight and three a.m.? Doesn't that wear you out emotionally? I'd try exercise, during the day, a whole lot of it for him (even if it just is running up and down while you count) and then soothing and calming in the night, in case the cause is emotional.
I would think that he is over tired by 9:30, and is not able to go into a deep sleep, that is why he is waking up. If he is over stimulated by the time he goes to bed, then his little mind is too busy to reach the deep sleep stage.
I would cut down his day naps and have him in bed by 7:30 asleep by 8:00. Kids that age need approx 10-12 hourse sleep for proper growth.
Make his bed time routine relaxing and quite, so no toys from 7:30, only a nice bed story, maybe some gentel music in the background. Try and get him as relaxed as possible. In the begninning it is going to be difficult to get him into the new idea of bed time, but it will pay off in the long run.
also try some warm milk, hunny and vanila at bed time - my son and call it his "bed time milk shake". it is natural relaxent - i even use it sometimes when i cant sleep.
I think that 9:30 is rather late for a 2 year old to be settling down to sleep at night on a regular basis - my oldest son is 4 and is in bed between 8 and 8:30 - on the very odd/rare occasion, he's been allowed to stay up later, but it's been due to a special circumstance - and, by then, I've found that he's just so overtired and crabby that it's just not worth allowing him to stay up later.
I agree with the previous poster who indicated that his daytime naps should be cut or limited. Every child is different, but my oldest son was ready to drop his nap altogether not long after he turned 2.
A bedtime routine is key and sets the stage for a good night's rest. A warm bath, a couple of stories and little to no stimulation. You may want to try something warm and soothing to drink at bedtime as the previous poster said, but this could also possibly disrupt him in the night by causing a wet diaper and some discomfort (even our 1 year old doesn't get anything else to drink after dinnertime when he has a sippy cup of milk).
All the best - I hope that your nights become calmer soon!! :)
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