  The thermodynamic and magnetocaloric properties of a generalized
spin-$(1/2,s)$ Fisher's super-exchange antiferromagnet are exactly investigated
by using the decoration-iteration mapping transformation. Besides the critical
temperature, sublattice magnetization, total magnetization, entropy and
specific heat, the isothermal entropy change and adiabatic temperature change
are rigorously calculated in order to examine cooling efficiency of the model
in a vicinity of the first- and second-order phase transitions. It is shown
that an enhanced inverse magnetocaloric effect occurs around the temperature
interval $T_c(B\neq 0) \lesssim T < T_c(B = 0)$ for any magnetic-field change
$\Delta B: 0 \to B$. The most pronounced inverse magnetocaloric effect can be
found nearby the critical field, which corresponds to the zero-temperature
phase transition from the long-range ordered ground state to the paramagnetic
one. The observed phenomenon increases with the increasing value of decorating
spins. Furthermore, sufficiently high values of decorating spins have also
linked to a possibility of observing reentrant phase transitions at finite
temperatures.
