Neither. It would most likely be locally distilled, or imported from the east coast. Distilling whisky was an important part of the American economy since before the union was formed. Enacting an excise tax on whisky production in 1791 led to one of the first crises of the newly formed country. http://www.mountvernon.org/research-collections/digital-encyclopedia/article/whiskey-rebellion/

It would likely have been corn whiskey, distilled in towns near corn farms in the Ohio River Valley (Ohio, Indiana, Illinois, Kentucky) and then a few decades later, the Midwest (with regards to corn: Iowa, Missouri, Nebraska, and Kansas). The population of these regions was increasing dramatically with farmers (often Scotch-Irish and English immigrants who brought with them whiskey distilling knowledge) in search of larger, cheaper, more fertile land.

The fertility of this soil is legendary. Glaciers retreating during the last Ice Age churned up the earth cycling nutrients to the top soil and it had never been planted (no erosion, no nutrient loss, full of nitrogen fixing bacteria, and humus -- the component in soil that holds moisture, much of which has been lost in American soil due to industrial, monoculture farming and heavy use of artificial fertilizers). The settlers farming this land knew none of this, but found that corn took extremely well to this soil. Corn soon became the chief cash crop of the frontier.

All this soil fertility led to a huge corn surplus. But there was no transcontinental railroad system, road network, or any other reliable way to send fresh, dried, or milled corn back to the population centers of the East Coast in economically viable quantities. There were two ways to create value using the surplus corn: pigs and whiskey. Both turned large quantities of corn into concentrated products that could be transported. Pork was salted, whiskey was barreled -- neither spoiled easily.

The surplus of corn made whiskey so cheap that it led to America's first public health crisis: mass alcoholism. Regular working people could now afford as much as a liter of whiskey per day. Fathers abandoned their families, domestic violence skyrocketed, people were drinking on the job (the concept of "Elevenses" was created in this era -- it refers to stopping work at 11AM to have a drink of whiskey, in addition to a breakfast, lunch, and dinner whiskey). Soon after, the temperance movement was born culminating in prohibition.

So this is the history that gave rise to Western saloons and the image of the whiskey-soaked cowboy. Corn is closely tied to many major developments in American history: it has a major role in the creation of cities like Chicago, Kansas City, St. Louis, and Cleveland, it allowed for America's first modern military mobilization (WWI), it allowed America to win the agricultural Cold War, it has made America a hegemon in processed food production, and on and on. Corn is up there as one of the most important products/inventions in American history.