You don't need new physics for interstellar spaceflight - 16 km/s of dV is enough. you don't even need to go that much faster to slowly spread among the stars. There are a lot of smaller bodies all the way from Sun to Alpha Centauri. As long as you hop between them within reasonable time in a few thousand years you can become a true interstellar civilization, while going at much-slower-than-light velocity (similar to Polynesian colonization of Pacific).