The Earth's magnetic field is very weak compared to that produced by a magnet you might '[slide] next to a coil' and the time over which the field will vary is long (> ~90 minutes for one orbit), so even in an optimally oriented coil the induced current would be very small. In the 'outer reaches of the solar system' any induced current would be orders of magnitude smaller still.
But in terms of using induced current as a power source there is a bigger problem - any energy extracted from the coil would be at the direct cost of the satellite's kinetic energy. Extraction of electrical energy would act like a (very inefective) brake on the satellite's motion.
As a postscript based on some of the comments below. In a uniform magnetic field a current loop (regardless of the magnitude of the current) produces exactly zero nett force, although it can produce a torque. So in deep space it could not produce any measurable acceleration or deceleration, as the sun's magnetic field is both very weak and almost uniform over any reasonable distance. For example: an enormous superconducting current loop with the same diameter as the earth (13,000 km), flying through space about half way between the orbits of Earth and Mars and carrying a current of one million amps, would experience a maximum force from the sun's magnetic field of approximately 20 Newtons. ...and that would drop to a milli-Newton about the time it passed Neptune's orbit.
