  CeRh$_2$As$_2$ is a multi-phase superconductor with
$T_{\textrm{c}}=0.26\,\textrm{K}$. The two superconducting (SC) phases, SC1 and
SC2, observed for a magnetic field $H$ parallel to the $c$ axis of the
tetragonal unit cell, have been interpreted as even- and odd-parity SC states,
separated by a phase boundary at $\mu_{\textrm{0}}H^{*}=4\,\textrm{T}$. Such
parity switching is possible due to a strong Rashba spin-orbit coupling at the
Ce sites located in locally non-centrosymmetric environments of the globally
centrosymmetric lattice. Existence of another ordered state (Phase I) below a
temperature $T_{\textrm{0}}\approx0.4\,\textrm{K}$ suggests an alternative
interpretation of the $H^{*}$ transition: It separates a mixed SC+I (SC1) and a
pure SC (SC2) state. Here, we present a detailed study of higher quality single
crystals of CeRh$_2$As$_2$, showing much sharper signatures at
$T_{\textrm{c}}=0.31\,\textrm{K}$ and $T_{\textrm{0}}=0.48\,\textrm{K}$. We
refine the $T$-$H$ phase diagram of CeRh$_2$As$_2$ and demonstrate that
$T_{0}(H)$ and $T_{\textrm{c}}(H)$ lines meet at
$\mu_{\textrm{0}}H\approx6\,\textrm{T}$, well above $H^{*}$, implying no
influence of Phase I on the SC phase switching. A basic analysis with the
Ginzburg-Landau theory indicates a weak competition between the two orders.
