The most plausible way to evade the speed of light is to exploit relativity itself, specifically, the time dilation effect. Time slows down for passengers on a spaceship approaching the speed of light. So a trip that appears from the outside to take thousands of years might seem to the passengers to take only a few months. Of course, when they returned back home, the space travelers would find their friends and family turned to dust, making this a not entirely satisfactory means of space travel. The latter problem figures prominently in the conclusion of Joe Haldeman’s Forever War, in which a soldier on an interstellar combat mission is separated from his girlfriend, and only a few years pass for him, while thousands of years pass back on Earth. (Haldeman employs a few additional tricks from relativity to ensure a happy ending). And the ultimate time-dilation tale is Poul Anderson’s Tau Zero. In Anderson’s novel, a colonization vessel begins to accelerate out of control. The time dilation becomes so extreme that the crew eventually witnesses (and survives!) the end of the universe itself.
A particularly heartrending exploitation of this idea is "The Old Equations," by Jake Kerr. (Do you know where the title of the story comes from?) Before I spoil it for you, you can read the story at Lightspeed magazine. Kerr's story actually pulls a clever swindle on the reader -- while it adheres slavishly to the laws of relativity, it introduces an idea that is just as implausible (in my mind) as violating the speed of light. But like a good magician, Kerr distracts your attention so that you don't even notice it.
"The Old Equations" postulates that Einstein died during World War I, so relativity never became an accepted part of physics. Time dilation is only discovered when an astronaut on the first interstellar mission finds it harder and harder to communicate with his wife back on earth. Eventually they realize that this is due to relativistic time dilation, with terrible consequences -- by the time the astronaut completes his mission, his wife will be long dead.
So what is Kerr's "swindle"? It's the idea that if Einstein had not survived, relativity never would have been discovered. Occasionally we scientists are guilty of the equivalent of the "Great Man" theory of history -- the idea that science is propelled forward by a handful of "Great Scientists," in whose absence the progress of science would have been radically stunted. This misconception is a natural result of the way that science is presented in popular accounts -- who isn't thrilled to read about Marie Curie toiling away to discover radium (and don't miss the movie version!) or Einstein, sealed away in a closed room, inventing relativity all by himself (of course that's not how it actually happened). But these are misunderstandings of the way that science really progresses. Science is highly collaborative -- if not directly, then through the influence that scientists have on each other. And for every "great discovery" by a single genius, there are often three or four other scientists hot on the same trail, or who even made the discovery first but didn't get the credit.
So if someone had waylaid Einstein, Fermi, Dirac, and Feynman, it would certainly have set back the progress of science, but only marginally -- others would taken their places. None of this, of course, takes away from "The Old Equations" itself -- it is, after all, science fiction.