The title alludes to the comic strip "Hairbreadth Harry", which ran from 1906 to 1940.
The last cartoon to pit Wile E. Coyote against Bugs Bunny, following Operation: Rabbit (1952), To Hare Is Human (1956), Rabbit's Feat (1960), and Compressed Hare (1961), and the only one in which Wile E. never speaks.
This was the first WB cartoon to be released after Warner Bros. Cartoons closed down in 1963.
The phone booth has a drawing of a misshapen bell on its wall, a likely nod to Bell Telephone, who provided most of the telecommunications to the US at the time.
Bugs says that the Road Runner has sprained a giblet, but that's not possible - a bird's giblets are its internal organs.