While Cabinet Secretaries get all the attention, there are over 4,000 politically appointed positions in the federal government. About a third of these are “Schedule C” appointments – mid-to-lower level support positions for more senior politically appointed individuals. Think policy advisers, counsels, personal assistants, schedulers, etc. Like other political appointees, they serve at the pleasure of the President and can be removed at any time.
However, there is the practice of “burrowing in” where these individuals will attempt to do an end-run around selective service requirements and convert their politically-appointed position into a civil service position. The Office of Personnel Management (OPM) is supposed to oversee and prevent this practice, but Trump has gutted OPM.