Your TSH rises because the thyroid is rebuilding its NISes. It needs the NIS because the NIS is the system that pumps Iodine
into the cell. When you're low on iodine, you loose your NISes. When you get (back) on iodine, the body rebuilds them. If you were really badly deficient for a long time, it can take months to rebuild them. The TSH stays high during that time.
Doesn't mean you're getting (more) hypo. TSH is a really lousy tool to measure the working of the thyroid by.
Here's the whole story: http://www.violiendamast.nl/thyroid
Other parts of your body use Iodine
too you know: quite a lot. So cells in others parts of your body are rebuilding NIS as well.
It will take a lot of time for the Iodine
to really start working. No NIS = no iodine in the cell. When I started I went from hyper to hypo a few times before it finally became a steady rise of well being.
Make sure your body is getting the co supplements. B2 and B3 are specifically needed to rebuild NISes. Also look at other deficiencies: in my experience iodine speeds up all your processes, which makes you need more nutrients. I was low in just about everything.
Warm hands will follow!