Most soldiers carried a night cap, which could often be a woolen watch cap or beanie.
The U.S. Army always slighted sun protection in favor of colder weather. In the 1820s the soldier was expected to wear an enormous, hot, and uncomfortable bell-crown shako with plumes and a metal shako plate and all sorts of frippery. Complaints about the impracticality led to the Model 1825 pin-wheel forage cap, modeled to some degree after Prussian and British antecedents. It was a big flat cap made of wool, sufficiently large to cover the face of a sleeping man. Also with a leather brim. Then came the Model 1833 leather folding forage cap, the infamous and unsightly "hog killer." The U.S. issued this out to soldiers fighting the Second Seminole War in Florida. Small wonder most soldiers in the south and southwest wore civilian straw hats and mufti while doing physical labor. The Model 1839 forage cap was made of wool and had ear flaps that could fold down to cover the nape of the neck. Hunting hats with Elmer Fudd type fur ear flaps are not regulation headgear, but they are certainly not new either.
The U.S. forage cap was made of wool, and if a scarf from home was tied around it, the ears would be taken care of.
By the 1870s muskrat fur caps were issued for severe winter weather, although informally such fur caps may have been worn long before on the frontier during serious cold weather. I'm not sure the army ever really made a very practical hat of any kind until the felt campaign hat...
At least lots of New Englanders, particularly New Hampshire, got whipple hats!