Won't the orbital path just become infinitely large with
increasing initial velocity?
No, is the short answer (more on that in a moment).
But firstly, your basic confusion appears to arise from mixing up two separate concepts:
a. For the orbital path to be infinitely large implies an infinite initial velocity (which obviously is not possible in any realistic scenario: nothing in the universe travels with infinite velocity).
b. For an orbital path to be possible at all implies a finite initial velocity -- and we're back to "no is the short answer" -- because:
i) If it was possible to have infinite initial velocity, by definition that must exceed the escape velocity, which is a defined, finite, velocity. The moment it does exceed that (finite) velocity, which would be long before reaching infinite velocity, any type of orbit becomes impossible: the rocket must go shooting off into space.
Orbit is an equilibrium state, where the force pushing the orbiting object away from the planet (basically, its momentum) is exactly equal to the force pulling it toward the planet (basically, gravity). If the outward impetus is in exact balance with the inward impetus, the equilibrium thus achieved causes the orbiting object to maintain a constant distance from the planet: an orbit.
ii) A slight imbalance in those forces will cause the orbiting object to orbit out to a greater radius distance, or to orbit in to a lesser radius distance; but a large (positive) imbalance will result in the object leaving orbit: it will cease to follow a curved path about the planet: the larger imbalance will cause it to follow, increasingly, a less curved path, until the path has largely ceased to curve at all: the object is now going in a straight line (approximately), so has ceased to orbit.
iii) Having once attained escape velocity, whether or not this follows upon an orbiting stage, if the rocket does no more than maintain its constant velocity it can never be recaptured by the planet's gravity, because the gravitational strength falls off with distance: under the inverse-square law, when the distance from the planet doubles the gravitational strength is not constant, but reduces to one-quarter. With merely constant speed, the rocket can never be captured by the falling gravitational force (because the latter is falling).
Only by decelerating could the rocket reduce its momentum such as to be travelling with less velocity than the critical threshold needed for equilibrium, i.e. an orbit, at its new radius distance from the planet (a very large deceleration would be involved, once the rocket has travelled even a short distance, because in real life, at a very short distance from the planet its gravity gives way to the Sun's far greater gravity, with the planet becoming thereafter a negligible influence).
So a further misunderstanding in your question now emerges: by treating the planet as if it was the only source of gravity in near-space, you have misled yourself into neglecting nearby stronger gravitational forces! Those forces overwhelm the planetary effects at quite short distances, such that your orbital-path assumptions break down, once the rocket has moved only quite a short distance from the planet.
The very small differences in orbital altitude associated with an eliptical orbit are irrelevent to my answer. The only relevent issue is whether the orbit is stable: if it is, minor altitude differences won't clarify the position any. I am addressing only the main issue: is the satellite-cum-rocket able to accelerate (taking it out of orbit), and is it able to decelerate (taking it out of orbit).
Like it or not, in reality all rockets burn fuel at a constant rate, and the only reason they go faster is that gravitational resistance to their motion decreases as altitude increases, obeying the inverse-square law.
Burning fuel at a constant rate, the amount of fuel it is possible to carry sets a real limit on its initial velocity (if it is going further than Earth orbit), and there are practical restrictions on the weight of fuel it is possible to lift (every extra ton of fuel in the tanks increases the mass which must be lifted into space). So increasing the payload of fuel has negative consequences also, not purely positive ones.
And a high enough initial velocity will prevent the rocket going into orbit: it will do what planetary probes do, and depart for Mars or beyond. If it never completes even a single orbit, it is not reasonable to describe it as having an orbit. If it will also depart the solar system, as some do, it will not even be orbiting the Sun, in any meaningful sense.
The gravitational influence of a single quark is tiny, for an individual quark, so it is only logical to conclude that its radius is also tiny. It is therefore not logical to conclude that its influence is infinite. The inverse-square law demonstrates that even with objects of stellar mass, the influence declines rapidly. Doubling the distance, the strength declines to a quarter, i.e. 25 percent (1 over 2 squared, i.e. 1/4). At 8 times the distance, the strength is only one percent (1 over 8 squared, i.e. 1/64th).