Doing some random searches, I learned that during the summer of 1946 several jeep races popped up across the country. One of them was very organized (#2), one was more of a performance (#1), one was impromptu (#4), and one provided no additional information (#3)

1. On the night of June 1, 1946, in Ogden, Utah, jeep races and jeep-jumps-jeep events were planned by the local American Legion, along with other event. The jeep events were part of Lt. Dick Ryan’s traveling jeep rodeo, so I’d argue they were more of a performance than an organic race. (Ogden Standard-Examiner, June 1, 1946).
2. A mid August 1946 five-mile jeep race for veterans was organized for the Plumas County Fair in California. Only standard jeeps and qualified WWII veterans could participate. Four jeeps were entered, with Clayt Joslin of Quincy  winning the event (As reported in the Feather River Bulletin August 22, 1946). First prize was $100, while second price was $50 (Indian Valley Record, Greenville, California, August 1, 1946).
3. Around September 14, an impromptu jeep race was held at the West Tennessee District Fair. It was organized by folks were were driving jeeps that were servicing the race track for the fair (The Jackson Sun September 15, 1946).
4. In October, jeep race(s) were held on October 04, 1946, at the Montana Youth Fair, in Kalispell, Montana. I could find no results or more information. (The Daily Inter Lake October 01, 1946, Kalispell, Montana)

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In unrelated news from 1953, this Yakima Ridgerunners photo hit some newspapers across the country. The example below was published in the June 24, 1953, issue of the Spokesman-Review, out of Spokane, Washington. Chet Thompson and Wally Klingele, both names synonymous with the early days of the Ridgerunners, are shown practice-racing for a July 4th event.