Even if we were to somehow detect a thousand more possible targets within that radius, I'm afraid we make no appreciable dent in the odds of finding an object on a viable trajectory. You are looking at a sphere with a surface area of 3142 square light years. So let's say the diameter of these targets is about 120 AU (a rough number for the radius of our heliosphere). That gives us an area of 0.000000215 square light years. That's a pretty tiny bullseye.

Although your comment about gravity assist implies that you are no longer interested in using the existing speed of the asteroid as you would need to manoeuvre it to slingshot around a gravity well, likely spending orders of magnitude more energy to do so than you have 'saved' by using an already moving object. At that point it would be be cheaper and easier to just build your space craft.