According to the 1961 statics on natality page 1-13, the rate of hospital births for whites was nearly 100% in 1961, having been about 60% in 1940, and rising sharply to 90% by 1948. Among non-whites the hospital rate climbed from 27% in 1940 to 86% in 1961.
Keep in mind that many of the 1940 home births actually had a physician in attendance.
According to the CDC, the percentage of out-of-hospital births has been rising since 2004 and is highest among whites (close to 2%).