Glad to hear of your good results and I agree with everyone in this thread that it is very important to keep your bowels moving.
A couple of different aids that you could take would be Oxypowder or Schulze intestinal formula # 1. I used both of those(not together) early on in my CZ days and I went from going once per day to going at least twice, and I've been that way for years now:). Both are easy on the system, the schulze a at a low dose(1 cap per day, taken in the evening) or the Oxypowder (elevate dosage to tolerance). These are both stop-gap measures though so be sure to not take either for an extended period of time.
A few thoughts. It could be that you have gone a bit to the hypo side as your body adjusts to the increased Iodine intake. You did mention though that constipation has always been a problem, so...
Diet and exercise... the basics:)
I'm sure you've heard the fiber proselytizing. Fiber's good, but excess fiber is not... as in baked goods with added bran, fiber this, fiber that, etc. Excess fiber in the diet and you actually run the risk of wiping out your good bacteria, leading to.... tada! constipation! Here, this dude says it better than I can:
So... my reco is to eat lots of fresh fruits and vegs, easy in the summer:) And good food for your friendly flora. Also be sure to get some fermented foods into your diet. Yogurt, kefir, buttermilk, kimchi, sauerkraut are all good... kombucha tea:)
BTW, in the old literature hypo has been cured by dietary means only. A specific diet that I've read of is one of only apples and buttermilk for.... 3 days? Makes sense. Fruit to feed the little buggers and bacteria in the buttermilk to repopulate.
Also, exercise. Ever notice what happens when people walk their dogs?
And, people used to take walks after dinner. That ritual was called a "constitutional". Exercise stimulates the bowels~!