  It has recently been suggested that at the post-inflationary stage of the
mixed Higgs-$R^2$ model of inflation efficient particle production can arise
from the tachyonic instability of the Higgs field. It might complete the
preheating of the Universe if appropriate conditions are satisfied, especially
in the Higgs-like regime. In this paper, we study this behavior in more depth,
including the conditions for occurrence, analytical estimates for the maximal
efficiency, and the necessary degree of fine-tuning among the model parameters
to complete preheating by this effect. We find that the parameter sets that
cause the most efficient tachyonic instabilities obey simple laws in both the
Higgs-like regime and the $R^2$-like regime, respectively. We then estimate the
efficiency of this instability. In particular, even in the deep $R^2$-like
regime with a small non-minimal coupling, this effect is strong enough to
complete preheating although a severe fine-tuning is required among the model
parameters. We also estimate how much fine-tuning is needed to complete
preheating by this effect. It is shown that the fine-tuning of parameters for
the sufficient particle production is at least $ < \mathcal{O}(0.1) $ in the
deep Higgs-like regime with a large scalaron mass, while it is more severe
$\sim {\cal O}(10^{-4})-{\cal O}(10^{-5})$ in the $R^2$-like regime with a
small non-minimal coupling.

High Energy Physics - Phenomenology;Cosmology and Nongalactic Astrophysics;General Relativity and Quantum Cosmology