Thinking outside the box here... how about the Lockheed CL-321 air tanker, which actually won the competition for the SAC jet tanker role filled by the Boeing 717, gets the bulk of USAF orders? The KC-135 was meant to be a stopgap whilst the Lockheed aircraft (which would probably have been K-1) was prepared for production. Assume this plan goes ahead. You then have 250 KC-135s and 550 K-1s, or thereabouts.
With the economies of scale from the USAF order, Lockheed's Constellation II - a commercial version of the CL-321 - is a very strong contender against the Boeing 707 and the Douglas DC-8. All three companies come up with short-range airliners to complement their long-range jets - Boeing with the 727 and 737, Douglas with the DC-9, and Lockheed with... something.
Later in the 1960s, the three four-jet airliners are more or less level pegging - the USAF orders for the Lockheed offering economies of scale that partly offset its' larger size and greater cost. Boeing's CX-HLS bid is successful, leading to the C-5 Stratolifter, whilst Lockheed's simple, low-risk L-2000 supersonic airliner gets the nod. Looking for a competitive advantage, Juan Trippe turns to Douglas - long their favoured supplier - to build a large airliner, capable of carrying freight should the supersonic transport turn out to be the expected titanic shift in air travel. Boeing and Lockheed rush to develop their own widebody airliners to compete, Boeing producing the widebody 747 trijet, whilst Lockheed produces the four-engined Galaxy, smaller than the big Douglas but with a longer range and cheaper to run.
That gives you Boeing taken down a peg whilst Lockheed and Douglas are playing in the big leagues, just as Airbus is entering the scene.