There are a wide variety of hats used during the 19th century. We see in our minds eye a top hat that Abraham Lincoln would use. The caps that those serving during the civil war wore and we even imagine the Stetson's hat from Texas fame (of course Stetson's shop was in Philadelphia, PA). Although he did invent the hat while in Colorado. Stetson retailed his hat business and by the end of the 19th century there were 150 whole merchants around the country.
Below is an excerpt from the Library of universal knowledge ©1880 and gives a brief description of how hats were made, specifically in the earlier half of the 19th century.
HAT, a well-known species of head-covering, which has assumed various characters. What we understand by a hat is a fabric of felt, or a silk material used as a substitute for felt. . .
The growing scarcity of beaver-fur led to attempts to substitute a cloth formed of silk plush, drawn over a pasteboard frame, about 1810. These were not very successful; and hats of wool or beaver-felt were common until about 1840. The high cost of beaver at length forced on the improvement of silk hats, and now the beaver is almost entirely superseded; while the fabrication of silk hats has been carried to great perfection not only in England, but in continental countries and the United States. The silk hat consist of a body and rim, usually made of two or three layers of cotton-cloth saturated with varnishes, to give the fabric stiffness, and make it waterproof. These are molded on wooden blocks according to the fashion of the day; and when the desired shape is produced, the whole is carefully furnished over with lac and dammar varnish, and, before dry, the fine silk plush is applied with great nicety, so as to prevent the seams being perceived; it is then trimmed with silk braid on the edge of the brim, and a silken band round the junction of the body with the brim; and the lining of leather and thin silk being put in, it is complete. Lightness, gloss, and durability are the prime qualities of the silk hat; and in these respects the hats of New-York manufacture deserve a high commendation. Very excellent hats are made in London, Paris, and Edinburgh; but they are heavier than those of America.