The New York Times recently featured an article on the original Filene's Basement store built in 1909, which showed a photograph taken of two men's suit racks. The date is uncertain but my best guess is that it dates back at least as early as the 1940s or 1950s.
Today it appears most mainstream American men's stores carry a jacket size range of 38 to 48 in even numbers (European equivalent is 48 to 58). A handful of specialty stores cater to smaller and larger sizes. What I found interesting is the photo shows jacket sizes starting at 33 and going sequentially to 34, 35 and 36 (and presumably increasing through the larger sizes as well). In other words, RTW jackets were offered in more precise sizing some fifty or sixty years ago.
I think I've only seen an odd-sized jacket in mainstream retail stores a handful of times. So it appears that a full range of even and odd sizes in RTW went out decades ago. If it exists today, I would guess the only retailers and makers who might go for this level of precision are the Japanese.