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.

The last study I saw estimated 7 interstellar system objects of Oumuamua's type per year https://arxiv.org/abs/2103.03289. Even if they were all suitable for conversion to a long term space craft, that's only 7 chances per year of an object on a trajectory that will intersect with something worth travelling to.

Oumuamua was fast relatively speaking. It was about 3 times as fast as the vehicle sent to the comet. Which means that Oumuamua was travelling at 0.0127% of the speed of light. So it will take 8000 years to travel 1 light year.

There are only 12 to 15 stars within a 10 light year radius of earth. So assuming we can build an 80,000 year craft out of our asteroid (when we can't build a dishwasher that lasts 5 (not a specific example, honest)), the chance of finding one on a suitable trajectory is effectively zero.

Aside from all the other points raised here the odds of finding any body travelling through our system that will also interersect another system in any non-geological time scale is infinitesimal. Space is too empty and extra-solar objects are too rare.