I imagine the answer will vary significantly by locale. In New Orleans, for example, liquor was still widely available, with waiters at some restaurants (e.g. Tujaque's, still an institution) carrying around small bottles in their aprons to discreetly add to patrons' drinks. Some of this was legitimately acquired booze, as there was an exemption in the Volstead Act for liquor to be used for "legitimate culinary purposes," e.g. sherry or brandy in soups or desserts. However, much of it was bootlegged, brought in via the port or down the river.

This is also a chance to tell my favorite New Orleans/Prohibition story:

Izzy Einstein was a highly successful federal agent of the U.S. Prohibition Unit who (between 1920 and 1925, using countless disguises to aid him in purchasing alcohol) made 4,932 arrests of bootleggers, bartenders and speakeasy owners. He found the illegal alcohol quickly: only twenty-one minutes in Chicago, seventeen in Atlanta, and in Pittsburgh a speedy eleven minutes. But in the Crescent City, it was an astonishing thirty-five seconds before an arrest. Hopping into a New Orleans taxi, Izzy asked the driver where he could get a drink. The taxi driver reached back and produced the goods almost immediately, whereupon Izzy cuffed him and broke a new speed record.

You can read more about Prohibition in New Orleans here: http://www.neworleansbar.org/uploads/files/Prohibition%20in%20New%20Orleans_1-8.pdf