In this work, we construct 36 tetraquark configurations for the $1S$-, $1P$-, and $2S$-wave states, and make a prediction of the mass spectrum for the tetraquark $ss\overline{s}\overline{s}$ system in the framework of a nonrelativistic potential quark model without the diquark-antidiquark approximation. The model parameters are well determined by our previous study of the strangeonium spectrum. We find that the resonances $f_0\left(2200\right)$ and $f_2\left(2340\right)$ may favor the assignments of ground states $T_{\left(ss\overline{s}\overline{s}\right)0^{++}}\left(2218\right)$ and $T_{\left(ss\overline{s}\overline{s}\right)2^{++}}\left(2378\right)$, respectively, and the newly observed $X\left(2500\right)$ at BESIII may be a candidate of the lowest mass $1P$-wave $0^{-+}$ state $T_{\left(ss\overline{s}\overline{s}\right)0^{-+}}\left(2481\right)$. Signals for the other $0^{++}$ ground state $T_{\left(ss\overline{s}\overline{s}\right)0^{++}}\left(2440\right)$ may also have been observed in the $\varphi \varphi$ invariant mass spectrum in $J/\psi \to \gamma \varphi \varphi$ at BESIII. The masses of the $J^{PC}=1^{--}$ $T_{ss\overline{s}\overline{s}}$ states are predicted to be in the range of $\sim 2.44–2.99\mathrm{GeV}$, which indicates that the $\varphi \left(2170\right)$ resonance may not be a good candidate of the $T_{ss\overline{s}\overline{s}}$ state. This study may provide a useful guidance for searching for the $T_{ss\overline{s}\overline{s}}$ states in experiments.