You can get a sunburn through a lightweight dress shirt.

I have found some of what you are looking for at thrift stores. Maybe look at Men's Warehouse or something similar?

When I was 16, my dad sent me off to Philmont with a wool sweater and one of his thin dress shirts. I don't think it was cotton. Amazing how well this combo works.

One other thing I thought of. If it has a low cotton amount, and it is really thin, the cotton will dry fairly quickly.
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