You might have come across "hacking jacket" as another term for sport coat.
A hacking jacket is a traditional English riding jacket, often "ratcatcher" (tweed). It's cut with a single vent to look good when sitting in the saddle, and has a tailored waist. Hacking jackets are sometimes now worn purely as fashion statements. In fact, Matt Smith's jacket as Doctor Who is close to a hacking jacket in style, especially the Shetland tweed one he wore last.