A bit of a necro post, but I can see you were trying to find the answer.
The
Strange Hood is the first piece you can find. For most players, running through the Ashen Cape to access the Sunken Citadel is the best bet. There's a Strange Hood hidden away. Getting the one from Bappasalar Cave first is totally doable, but you might find the dungeon tougher than the Sunken Citadel. Plus, the citadel drops some nifty loot.
The
Googled Helmet only drops in the last dungeon of the main quest.
Note, while having Anti-radiation is useful in radiation areas, it's not strictly required to beat the game. You can simply run through the Ashen Cape, and the other area that has the hazard. (Being vague to avoid spoilers). But, yeah, it's useful if you plan on fighting in those areas.