  This paper points out that the full-orbit density obtained in the standard
electrostatic gyrokinetic model is not truly accurate at the order
$\varepsilon^{\sigma-1}$ with respect to the equilibrium distribution
$e^{-\alpha \mu}$ with $\mu \in (0, \mu_{\max})$, where $\varepsilon$ is the
order of the normalized Larmor radius, $\varepsilon^{\sigma}$ the order of the
amplitude of the normalized electrostatic potential, and $\alpha$ a factor of
$O(1)$. This error makes the exact order of the full-orbit density not
consistent with that of the approximation of the full-orbit distribution
function. By implementing a hybrid coordinate frame to get the full-orbit
distribution, specifically, by replacing the magnetic moment on the full-orbit
coordinate frame with the one on the gyrocenter coordinate frame to derive the
full-orbit distribution transformed from the gyrocenter distribution, it's
proved that the full-orbit density can be approximated with the exact order
being $\varepsilon^{\sigma-1}$. The numerical comparison between the new
gyrokinetic model and the standard one was carried out using Selalib code for
an initial distribution proportional to $\exp(\frac{-\mu B}{T_i})$ in constant
cylindrical magnetic field configuration with the existence of electrostatic
perturbations. In such a configuration, the simulation results exhibit similar
performance between the two models.
