Various Types of Hay

Naturally, where your story is set has a bearing on which kind of hay you would us but here's a short list of some of the types.

Clover hay has a higher feeding value ton for ton than meadow hay or corn-fodder. It is so much superior that I must be pardoned for illustrating it from Stewart's tables.
Source: The Breeder's Gazette ©1895
(I didn't include the table for this tidbit.)

Timothy hay is almost universally considered as the best of the long foods for horses. yet many hays from mixed grasses are used. and is some sections alfalfa hay. In recent years in some sections cut and shredded corn fodder has become very popular. and for many years corn blades have been preferred. in the South. by the keepers of race horses.

I prefer Orchard grass hay to timothy hay as it has more blades, timothy dies out in the course of a few years, while an Orchard grass sod will continue to get better each year for many years. One acre of Orchard grass will afford as much pasture as two of clover and timothy. I believe timothy to be an impoverisher of the land, while Orchard grass forms such an immense sod that for plowing under it is equal to a clover one.
Source Henderson's Handbook ©18 quote came from a man in VA.

Alfalfa hay is preferable to either clover or timothy for farm animals, and especially for swine, one acre being worth three of clover for hogs. It is also good for horses, and for oatile it is worth three times as much as red clover.
Source: Report of Kansas State Board of Agriculture ©1893

Oat Hay The results of the experiments indicate that the nutrients of oat hay are in the most digestible form when the heads are in milk. If cut in bloom there is a less yield of poorer composition and digestibility than when cut in milk. If the cutting is delayed till the oats are in the dough stage, the slightly larger yield is more than offset by the poor quality and lessened digestibility of the hay.
Source: Annual Report of Maine ©1898

Below is a list without descriptions of various hays:
Meadow Fescue Hay
Mountain Rye Grass Hay
Canary Reed Grass Hay
Salt Grass Hay
Lupine
White Lupine
Wild Oats Hay
Wheat Hay
Red Top Hay