Hitting escape velocity won't help -- whatever is sent out will be unrecoverable. It will roughly be on Earth's orbit around the Sun, but likely won't coincide.

There is some flexibility by shooting the object past the moon, and getting a little bit of a 'slingshot' effect to modify the resulting orbit. But it would still require steering rockets because even a slight error in the initial path or orbital calculation would make a huge different in the resulting orbit.

This topic would make for a great physics and geometry lesson. Why inertial orbits starting from the surface always intersect the surface (the original topic). Why going from a one-body system to a two-body system might be able to change this, but at the expense of extreme sensitivity. How the complexity of potential orbits vastly increases as you add more bodies.

Even more interesting, is the extreme state sensitivity of the "interesting" configurations. A few seconds of arc difference in the initial course can put you someplace completely different. Simple orbital calculations assume point sources of gravity. That's not a bad approximation if you are far enough away, but a "slingshot" breaks that assumption. You can't even model the objects as uniform spheres -- the earth isn't spherical, and it doesn't have a uniform mass distribution. And for objects such as the Earth that have significant magnetic fields, there will be a deflection on approach and departure.

Bottom line is that anything done in space requires a significant ability to steer, and most operations require the ability to dynamically navigate.