  UTe$_2$ is a recently discovered promising candidate for a spin-triplet
superconductor. In contrast to conventional spin-singlet superconductivity,
spin-triplet superconductivity possesses spin and angular momentum degrees of
freedom. To detect these degrees of freedom and obtain the solid evidence of
spin-triplet superconductivity in UTe$_2$, we performed $^{125}$Te-NMR
measurement. We previously reported that the shoulder signal appears in NMR
spectra below the superconducting (SC) transition temperature $T_{\rm c}$ in $H
\parallel b$, and that a slight decrease in the Knight shift along the $b$ and
$c$ axes ($K_b$ and $K_c$, respectively) below $T_{\rm c}$ at a low magnetic
field $H$. To clarify the origin of the shoulder signal and the trace of the
decrease in $K_b$, we compared the $^{125}$Te-NMR spectra obtained when
$H~\parallel~b$ and $H~\parallel~c$ and measured the $^{125}$Te-NMR spectra for
$H~\parallel~b$ up to 14.5~T. The intensity of the shoulder signal observed for
$H~\parallel~b$ has a maximum at $\sim 6$~T and vanishes above 10~T, although
the superconductivity is confirmed by the $\chi_{\rm AC}$ measurements, which
can survive up to 14.5~T (maximum $H$ in the present measurement). Moreover,
the decrease in $K_b$ in the SC state starts to be small around 7~T and almost
zero at 12.5~T. This indicates that the SC spin state gradually changes with
the application of $H$. Meanwhile, in $H~\parallel~c$, unexpected broadening
without the shoulder signals was observed below $T_{\rm c}$ at 1~T, and this
broadening was quickly suppressed with increasing $H$. We construct the
$H$--$T$ phase diagram for $H~\parallel~b$ and $H~\parallel~c$ based on the NMR
measurements and discuss possible SC states with the theoretical consideration.
We suggest that the inhomogeneous SC state characterized by the broadening of
the NMR spectrum originates from the spin degrees of freedom.
