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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.