An object in orbit follows an elliptical path (a circular orbit is just a special case of an elliptical orbit).

When a satellite approaches escape velocity, the ellipse becomes very "stretched": long and skinny (high eccentricity).

At the farthest point in the orbit, the satellite is moving most slowly. So, just below escape velocity, the satellite would be almost stationary, and could spend millions of years far from the Sun. You would need to measure it's velocity very carefully to see if it would fall back in towards the Sun in an elliptical orbit, or if it would drift on "to infinity".

Additional information:

If the satellite escapes to infinity, it is not called

an orbit, because it is no longer a closed path.

If the satellite has

*exactly* escape velocity, it's path will follow (part of) a parabola.

If the satellite has

*more than* escape velocity, it's path will follow a hyperbola.

The escape velocity is usually measured from ground level - if an object

*already *in orbit is then accelerated to just below escape velocity (for the ground), then it

*will *escape to infinity. Such an path is an hyperbola.

Circles, Ellipses, Parabolas & Hyperbolas are all

conic sections, and there is a very small velocity separating an elongated ellipse (eccentricity→∞) from a parabola or a hyperbola.