  We report the specific heat of single crystals of the spin ice compound
Dy$_2$Ti$_2$O$_7$ at temperatures down to 100 mK in the so-called Kagome ice
state. In our previous paper, we showed the anisotropic release of residual
entropy in different magnetic field directions and reported new residual
entropy associated with spin frustration in the Kagome slab for field in the
[111] direction. In this paper, we confirm the first-order phase transition
line in the field-temperature phase diagram and the presence of a critical
point at (0.98 T, 400 mK), previously reported from the magnetization and
specific-heat data. We newly found another peak in the specific heat at 1.25 T
below 0.3 K. One possible explanation for the state between 1 T and 1.25 T is
the coexistence of states with different spin configurations including the 2-in
2-out one (Kagome ice state), the 1-in 3-out state (ordered state) and
paramagnetic one (free-spin state).
