Rabosai: When you protect your hat from the rain with a plastic shopping bag, especially a plastic shopping bag which promotes the local market, and then walk through the streets wearing the hat cum bag on your head YOU LOOK STUPID.
A hat is worn to keep your head warm and/or dry. During wet weather, cover your head with a rain hat. Don't wear a hat that needs to be covered itself. If you're the sort who won't pray without a hat on your head, wear the rain hat into shul, and daven with that on your head. No one will object. In a room full of men, odds are no one will notice. And here's another secret: If you do wear a rain hat and choose to leave it in the cloak room, no one will think bad of you if you for davening hatless. They aren't required.
Back when I was a wee lad, the old timers in our shul all saved their felt Barcelenos and Stetsons for better weather, and wore London Fog rain hats when it was wet. To the best of my knowledge, all of their sons were admitted to fine yeshivas, and all of their daughters were wed to estimable men. My own father-in-law, an estimable man in his own right, goes without a hat, and his daughter did pretty well, don't you think? So, please, Yeshiva Boys, put aside your superstitious fears. Wearing an ordinary rain hat, instead of a bag-covered hat in the rain carries no consequences. (Except the very desirous consequence of NOT LOOKING STUPID.
Incidently, this new habit of treating an ordinary hat as if it were a sacred ritual item is why the hat is one of the things I like least about Orthodox Judaism.