I know that musicians often traveled around to wherever they could (traveling minstrels) but would say, a saloon/pub in Georgia be like "I want john Smith from NYC!" Did patrons ever request specific musicians to their local bar owner?
Edit: As pointed out by a mod, I want to narrow is down to specifically the post Civil War period to the rough period when radio was introduced and widely used.
Can you specify a general place or time period? Right before the advent of consumer recorded music in America perhaps?
I'm not sure what you mean by 'gigging,' but there were absolutely 'in-demand' musicians in 1860s-1890s America. Ole Bull, to name one, was a famous violinist and heavy tourer. Bull is a good example of someone who straddled 'classical' and 'popular' music as we now understand them--he played songs by classical composers, but also folk tunes, and attracted a wider range of audience than would have attended, say, a big-city opera house. Opera singers, of course, had well-known reputations and went in the road in touring productions, but I suspect they're outside what you mean by 'gigging' musicians. There were also traveling vaudeville/circus/minstrel acts which incorporated music and had wide public reputations.