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.
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