  The tetragonal intermetallic compound EuAl$_{4}$ hosts an exciting variety of
low temperature phases. In addition to a charge density wave below 140 K, four
ordered magnetic phases are observed below 15.4 K. Recently, a skyrmion phase
was proposed based on Hall effect measurements under a $c$-axis magnetic field.
We present a detailed investigation of the phase transitions in EuAl$_{4}$
under $c$-axis magnetic field. Our dilatometry, heat capacity, DC magnetometry,
AC magnetic susceptibility, and resonant ultrasound spectroscopy measurements
reveal three magnetic phase transitions not previously reported. We discuss
what our results reveal about the character of the magnetic phases. Our first
key result is a detailed $H \parallel [001]$ magnetic phase diagram mapping the
seven phases we observe. Second, we identify a new high-field phase, phase VII,
which directly corresponds to the region were skyrmions have been suggested.
Our results provide guidance for future studies exploring the complex magnetic
interactions and spin structures in EuAl$_{4}$.
