Assumptions
At 1 percent of the speed of light, it would take over 400 years to get to Alpha Centauri. That takes a LOT of dedication, and assumes that there are places at your destination to create a civilization to spread even further.
Terraforming local planets, moving into space and colonizing asteroids, building O'Neill colonies, or even Dyson Spheres or Ringworlds, might all look a lot more attractive for elbow room and resources than going out and finding a lot of new, livable solar systems at 1 percent of the speed of light. I could see some local colonization being undertaken, especially if instruments detected livable planets.