I don't think Rodgers and Hammerstein have had another No. 1 single in the UK or the US besides the two already mentioned.
Perry Como's cover version of Rodgers and Hammerstein's "Some Enchanted Evening" was No. 1 in the US for five weeks in 1949. The Billboard Hot Hundred chart didn't exist under that name until 1958, but in the 1940s they published three different song popularity charts, one of which was Best-Selling Singles.
