a) you give it enough energy to reach escape velocity, in which case there's no problem.

b) you blow out the circular orbit into a long ellipse. The far end, apogee, can be very far out depending on how much energy you added, but the near end, perigee, is always at the same altitude as it was at the moment of impact. Therefore, as long as the satellite is "within" thin atmosphere at the moment of impact, the fragments will periodically pass through thin atmosphere as well.

To properly boost yourself into a higher (circular) orbit you need to give yourself two kicks: one to raise your apogee, and then half an orbit later another to raise your perigee to match.