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The present measurement of the standard model (SM) parameters suggests that the Higgs effective potential has a maximum at the intermediate scale, and the electroweak (EW) vacuum is not absolutely stable. The simplest possibility for absolute EW stability may be introducing a very large Higgs-Ricci scalar non-minimal coupling. In this extension of the SM, I study the cosmic inflation driven by the Higgs field. Since the resulting Hubble parameter happens to be around the weak scale, I call this scenario $weak\text{-}scale\text{ }Higgs\text{ }inflation$. It is pointed out that the cosmic-microwave background normalization of the scalar density perturbation can be enhanced compared with the conventional Higgs inflation prediction, thanks to the potential maximum predicted by the SM. It turns out in the Palatini formulation of the gravity with various higher dimensional terms suppressed by the scale close to the Planck scale that the inflation is successful, with a prediction of the running of the spectral index, $α_s=-(3-4)\times 10^{-3}$, which can be tested in the future. I also argue the UV (in)sensitivity of the prediction, the fact that the parameter region for successful inflation is very close to the criticality for potential shapes for eternal and non-eternal inflation, and the phenomenological applications.