Well, now. If we were to throw a conveniently-sized black hole through the Earth-Moon gap during the rocket's transit, could it work to give them a tow, or at least a sufficient trajectory change?
/me reaches for a convenient envelope-back
(We will ignore the resulting mile-high tide as convenient sequel fodder.)
@cerebrate @nyrath @TheSpaceshipper
Yes, it's absolutely possible for a freak encounter with a rogue black hole to produce the required trajectory change.
Basically, a gravity assist maneuver can roughly provide a trajectory change up to twice the escape velocity of the black hole. Obviously a black hole has a ridiculously high escape velocity at its event horizon, but for story purposes we're limited to the escape velocity above where "spaghettification" becomes a problem.