  Exoplanet imaging and spectroscopy are now routinely achieved by dedicated
instruments on large ground-based observatories (e.g. Gemini/GPI, VLT/SPHERE,
or Subaru/SCExAO). In addition to extreme adaptive optics (ExAO) and
post-processing methods, these facilities make use of the most advanced
coronagraphs to suppress light of an observed star and enable the observation
of circumstellar environments. The Apodized Pupil Lyot Coronagraph (APLC) is
one of the leading coronagraphic baseline in the current generation of
instruments. This concept combines a pupil apodization, an opaque focal plane
mask (FPM), and a Lyot stop. APLC can be optimized for a range of applications
and designs exist for on-axis segmented aperture telescopes at $10^{10}$
contrast in broadband light. In this communication, we propose novel designs to
push the limits of this concept further by modifying the nature of the FPM from
its standard opaque mask to a smaller size occulting spot surrounded by
circular phase shifting zones. We present the formalism of this new concept
which solutions find two possible applications: 1) upgrades for the current
generation of ExAO coronagraphs since these solutions remain compatible with
the existing designs and will provide better inner working angle, contrast and
throughput, and 2) coronagraphy at $10^{10}$ contrast for future flagship
missions such as LUVOIR, with the goal to increase the throughput of the
existing designs for the observation of Earth-like planets around nearby stars.
