Everyone owns cotton or NYCO blend items, including hats.  Perhaps some more-or-less pure cotton GI  "overwhites"  This thread is NOT intended to transform such garments into "water-proof" items, but to add some "water-resistance" to them.

First thing to do is to launder the garments, using a suitable detergent.  I suggest basic Woolite, selected because it has no "optical brighteners", and because it is easily available.

Launder your garments, appropriately, using the selected detergent.

Here's the tricky part, and please allow me to expand;  This expansion is for the benefit of people who be ignorant of such things.  Not stupid, just uninformed.  To the extent your garment has a cotton content, such cotton will shrink when exposed to high temps.  The higher the cotton content, the more the shrinkage.  Best case, hand laundering/treating with warm water.  Worst case washing in hot water, and then drying in a clothes dryer.  The higher the cotton content, the worse the shrinkage.

Some mfrs will mention that their garments are made from "pre-shrunk" cotton.  Take this with a grain of salt.

I have found that repeated hand-launderings in warm water lead to successive shrinking of the item, the more cotton, the more the shrinkage, until a point where the item would not further shrink.  Unfortunately, seems inevitable.

So what to do?  Buy your pure cotton garments a bit larger than your "usual" size, maybe quite a bit.  Consider buying your cotton/nylon blend garments up to the next size, particularly if you are close to a "sizing" break, or the higher the cotton content.  Maybe a full size larger, in all dimensions.  Another example:  I once bought a cotton jacket which shrank a lot, after hi-temp maltreatment.  The worse part was not the chest, which shrank a little, but the arms, which seemed to shrink a LOT.  Unwearable after that mistake.

More later; running out of battery power.....

ETA, and Thanks for your patience!

The rest is simple enough, but like any task, prior prep prevents poor performance.  Just wash the selected garments in "Cotton Proof" available from many vendors.  The stuff I found is made by Nikwax.  Let air-dry.  Done.

This will NOT make your cotton garments waterproof, but will certainly help some cotton outerwear withstand light rain.  I believe it's best used in cotton hats and jackets typically worn in warm weather.  YMMV.

Disclaimer: no financial interest in any vendor/mfr.

Just treated a number of pure cotton, and NYCO blend, hats/caps. Need to await their drying.  Will report results.

The Mfr's suggestion on wearing rubber gloves is a good one, if hand-laundering.

ETA: complete success.  My cotton/NYCO hats, which formerly absorbed a light rain, and kept the absorbed moisture with me for a long time, now shed water like a duck's back.  No telling how long the water-resistance will last, but I reckon laundering in detergents intended for such purposes, will help.  Even so, for anything other than a "work" hat/coat/pants which will likely get filthy quickly, this is a viable option for most other hats/outer garments, and a pleasant outcome.

Submitted for consideration.

No change in the feel of the fabric?

Not that I can detect.