  Reentrant cavities are microwave resonant devices employed in a number of
different areas of physics. They are appealing due to their simple frequency
tuning mechanism, which offers large tuning ranges. Reentrant cavities are, in
essence, 3D lumped LC circuits consisting of a conducting central post embedded
in a resonant cavity. The lowest order reentrant mode (which transforms from
the $TM_{010}$ mode) has been extensively studied in past publications. In this
work we show the existence of higher order reentrant post modes (which
transform from the $TM_{01n}$ mode family). We characterize these new modes in
terms of their frequency tuning, filling factors and quality factors, as well
as discuss some possible applications of these modes in fundamental physics
tests. The appendix contains a comment on a paper related to this work.
