First off, it's important to note that there's a very important point about reference frame here. Right now, just standing on the Earth's surface, you are moving on a path that is, to a first order, a (slightly) elliptical orbit around the Sun, because the Earth is carrying you along in its orbit.
If you then climb atop a suitably good rocket and launch yourself to an escape trajectory from Earth, you'll now be moving on what amounts to a modest perturbation of this orbit whose exact shape will depend on the direction of launch and any excess speed in the case where there was more fuel than the minimum needed and the engine was allowed to burn past that point.
However, from the reference frame on the Earth, then of course, you are at rest (presumably, unless moving in some other way while reading this). When you launch off to your escape course, this is the reference frame where a hyperbolic trajectory is seen - up until a point.
I bring up this point about reference frame because it's necessary for the next part, which is in ascertaining where that "point" actually is (it's not actually an exact location, but rather a transition zone), as it has to do with the balance of forces affecting your motion.
Since Earth's core (not surface; we don't want to consider the rotation around the axis for this purpose!) is not an inertial reference frame, the force description must include a "fictitious" (i.e. not generated by an interaction) centrifugal term, along with the gravitational forces due to both the Earth and the Sun (let's ignore the Moon for now, that will just complicate [if not 'terminate with extreme prejudice' :D] the trajectory even more if you have an encounter - you see what's going on here? - and assume the coast is clear under the launch circumstances).
The description of the escape as following the hyperbolic course, then, is valid so long as the assumption under which it's made - which is basically that the sole source of gravity is the Earth, and it is a perfectly symmetric gravitator - is valid. In this case, the latter part is not so important as the former: the model will fail once the gravitational force from Earth is no longer large compared to the sum of these other two forces.