  We apply the QCD sum rule method to systematically study the $S$- and
$P$-wave fully-strange tetraquark states within the diquark-antidiquark
picture. We systematically construct their interpolating currents by explicitly
adding the covariant derivative operator. Our results suggest that the
$f_0(2100)$, $X(2063)$, and $f_2(2010)$ may be explained as the $S$-wave $s s
\bar s \bar s$ tetraquark states with the quantum numbers $J^{PC} = 0^{++}$,
$1^{+-}$, and $2^{++}$, respectively. Our results also suggest that both the
$X(2370)$ and $X(2500)$ may be explained as the $P$-wave $s s \bar s \bar s$
tetraquark states of $J^{PC} = 0^{-+}$, and both the $\phi(2170)$ and $X(2400)$
may be explained as the $P$-wave $s s \bar s \bar s$ tetraquark states of
$J^{PC} = 1^{--}$. The masses of the $s s \bar s \bar s$ tetraquark states with
the exotic quantum number $J^{PC} = 1^{-+}$ are extracted from two
non-correlated currents to be $2.45^{+0.20}_{-0.25}$ GeV and
$2.49^{+0.21}_{-0.25}$ GeV.
