When the wolf looks through the window and sees the little pig looking up the chimney, he is shown imagining the little pig as a pair of hams. A sign stuck into one of the hams reads "48 points". This indicates the number of red food rationing points to purchase this item, as this film was made and released during World War II. 48 points was a lot, equaling three-fourths of a person's red points (for meat, fish and dairy) for a month.
The piglet is based on Red Skelton's Mean Widdle Kid. The wolf (especially when he laughs) recalls Harold Peary's radio (and later film) character, The Great Gildersleeve (1942). The Father pig is based on Andy Devine.
10 years earlier, Pinto Colvig played the same part (the pig who built his house from bricks) in the Disney classic, "Three Little Pigs."
The title - "One Ham's Family" - is a parody of the long-running radio soap opera (1932-1959) called "One Man's Family."
Showing a married couple sleeping in the same bed, even a married cartoon pig couple, was just this side of the censors in 1943. Human couples in film were largely resigned to twin beds.