The only other real competitor was the VC-10, and it failed even more miserably to compete with the 707 than the Convair planes did.
The problem is that the 707's position wasn't gained on any real technical merits, but because Boeing was by far the most responsive of any airline maker in terms of making design changes the customers wanted. All its rivals did something better - the VC-10 had better takeoff performance and was more comfortable; the Super 60 DC-8s had greater capacity for far fewer design headaches; and the Convairs 880 and 990 were faster. But Boeing listened to their customers and were willing to tailor the 707 to their specific needs.
That said, the Super 60 DC-8s, not to mention the later 747, offer a tantalizing opportunity: compete on size. The Super 60s are by far the most successful 707 competitors, because their capacity for 200+ seats was something the 707 couldn't match. Further, the Super 60 series is responsible for the DC-8 outlasting the 707 on the cargo market.
The VC-10 had an opportunity to do so - they almost had a variant of such size from the start - but British Airways was convinced they'd never be able to fill 200 seats. Which, of course, points to a bit of a chicken and egg problem with this idea. Namely, that the best way to compete with the 707 looks non-viable until the 707 causes an explosion in air travel. And by then, the 747 looms and just blows all narrow-body competitors out of the water.