  A reheating phase in the early universe is an essential part of all
inflationary models during which not only the Standard Model (SM) quanta are
produced but it can also shed light on the production of dark matter. In this
work, we explore a class of reheating models where the reheating is induced by
a cubic interaction of the inflaton $\phi$ to the SM Higgs boson ${\textbf h}$
of the form $g_{h\phi} M_{\rm Pl} \phi |{\textbf h}|^2$ adopting the
$\alpha$-attractor T-model of inflation. Assuming inflaton as a background
field such interaction implies a $\phi$-dependent mass term of the Higgs boson
and a non-trivial phase-space suppression of the reheating efficiency. As a
consequence, the reheating is prolonged and the maximal temperature of the SM
thermal bath is reduced. In particular, due to oscillations of the inflaton
field the $\phi$-dependent Higgs boson mass results in periodic transitions
between phases of broken and unbroken electroweak gauge symmetry. The
consequences of these rapid phase transitions have been studied in detail. A
purely gravitational reheating mechanism in the presence of the inflaton
background, i.e., for $g_{h\phi}=0$, has also been investigated. It turned out
that even though it may account for the total production of SM radiation in the
absence of $g_{h\phi}$, its contribution to the reheating is subdominant for
the range of $g_{h\phi}$ considered in this work. As a dark matter candidate, a
massive Abelian vector boson, $X_\mu$, has been considered. Various production
mechanisms of $X_\mu$ have been discussed including (i) purely gravitational
production from the inflaton background, (ii) gravitational freeze-in from the
SM quanta, (iii) inflaton decay through a dim-5 effective operator, and (iv)
Higgs portal freeze-in and Higgs decay through a dim-6 effective operator.
Parameters that properly describe the observed relic abundance have been
determined.
