What was the purpose of the tiny rooms surrounding the Heichal?
I've never seen anything about how they were used, but there are a lot of them and they are in a very holy place.
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My best guess is they were primarily used for storage. The verses in Tanach often refer to the otzaros of the Beis Hashem without saying where they were. These rooms seem to be good candidates.
I also found this answer in a sefer in my house called Beis Hamikdash Harishon in the name of a sefer Tavnis Heichal.