  Taking advantage of Gaussian process (GP), we obtain an improved estimate of
the Hubble constant, $H_0=70.41\pm1.58$ km s$^{-1}$ Mpc$^{-1}$, using Hubble
parameter [$H(z)$] from cosmic chronometers (CCH) and expansion rate function
[$E(z)$], extracted from type Ia supernovae, data. We also use CCH data,
including the ones with full covariance matrix, and $E(z)$ data to obtain a
determination of $H_0=72.34_{-1.92}^{+1.90}$ km s$^{-1}$ Mpc$^{-1}$, which
implies that the involvement of full covariance matrix results in higher values
and uncertainties of $H_0$. These results are higher than those obtained by
directly reconstructing CCH data with GP. In order to estimate the potential of
future CCH data, we simulate two sets of $H(z)$ data and use them to constrain
$H_0$ by either using GP reconstruction or fitting them with $E(z)$ data. We
find that simulated $H(z)$ data alleviate $H_0$ tension by pushing $H_0$ values
higher towards $\sim70$ km s$^{-1}$ Mpc$^{-1}$. We also find that joint $H(z)$
+ $E(z)$ data favor higher values of $H_0$, which is also confirmed by
constraining $H_0$ in the flat concordance model and 2-order Taylor expansion
of $H(z)$. In summary, we conclude that more and better-quality CCH data as
well as $E(z)$ data can provide a new and useful perspective on resolving $H_0$
tension.
