  We systematically calculate the mass spectrum of the higher excited
fully-charmed tetraquark $cc\bar c\bar c$ states including the S-wave radial
excitations and the P-wave states with a nonrelativistic quark model. The quark
model is composed of a vector one-gluon-exchange (OGE) and a scalar linear
confinement interaction with the parameters determined by the charmonium
spectrum. In the calculation, we consider both the $3_c-\bar 3_c$ and the
$6_c-\bar{6}_c$ color representations. For the $cc\bar c\bar c $ state, the
$6_c-\bar 6_c$ component is located lower than the $\bar 3_c-3_c$ one because
of the stronger attractive interactions between the diquark and antidiquark. We
focus on two excited modes and their properties for the P-wave tetraquarks. The
mass splitting for the $\rho$-mode excitations with different color
configurations is large. The low-lying $6_c-\bar 6_c$ $\rho$-mode component
helps to explain the small mass gap between the ground S-wave and the P-wave
tetraquark states. The recently observed $X(6900)$ state may be the candidate
of the first radially excited tetraquarks with $J^{PC}=0^{++}$ or $2^{++}$, or
the $1^{+-}$ or $2^{-+}$ P-wave states based on the mass spectrum. Moreover,
the lowest $T_c$ states with the exotic $J^{PC}$ quantum numbers $0^{--}$ and
$1^{-+}$ may decay into the P-wave $\eta_c J/\psi$ and di-$J/\psi$ modes,
respectively. The future experimental search of these $T_c$ states will enrich
the hadronic spectrum.
