[Symmetry]

Are you thinking of comets? Asteroids are generally closish to the ecliptic. You'd normally want to find easy to reach asteroids rather than randomly choosing asteroids and there are a number you can get to for just 100 m/s beyond Earth escape, though it'll take a while at that speed.

The problem is that nearby asteroids tend to be dry since they're being warmed by the Sun as much as Earth is and lack Earth's gravity well, ozone layer, and magnetic field. You want to go to Jupiter's trojans or so, or maybe just Ceres if you're willing to deal with its gravity well and dig a bit. For Jupiter's orbit you're looking at 3,300 m/s plus escape. Still a lot less than getting to orbit and you can use an ion drive but hardly trivial.