Until 1932 neither 3 miles nor 5k were standard events in the major English-speaking countries. The AAU championships had the 2-mile and 5-mile while the British AAA had a 4-mile and 10-mile.
Here's your 3-mile AR--not official but as close as I can figure out from various sources at hand, from 1921 through 1950. Oddly enough they were all indoors in Madison Square Garden. Time in parenthesis is an approximate 5k equivalent; those in bold are superior to the then-current 5k AR.
14:37.6i (15:11.8) Earle Johnson -- New York Feb 1 1922
14:16.0i (14:49.4) Joie "Chesty" Ray -- New York Feb 28 1923
14:14.8i (en route to 14:39.0) Don Lash -- New York Feb 26 1938
14:05.0i (en route to 14:30.9) Don Lash -- New York Feb 25 1939
13:52.3i (14:24.8) Greg Rice -- New York March 31 1940
13:51.0i (14:23.4) Greg Rice -- New York Feb 22 1941
13:45.7i (14:17.9) Greg Rice -- New York feb 28 1942
Rice's best outdoor time was 14:00.0 in 1942.