  In this work, we investigate the nature of the fractional quantum Hall state
in the 1/3-filled second Landau level (SLL) at filling factor $\nu=7/3$ (and
8/3 in the presence of the particle-hole symmetry) via exact diagonalization in
both torus and spherical geometries. Specifically, we compute the overlap
between the exact 7/3 ground state and various competing states including (i)
the Laughlin state, (ii) the fermionic Haffnian state, (iii) the
antisymmetrized product state of two composite fermion seas at 1/6 filling, and
(iv) the particle-hole (PH) conjugate of the $Z_4$ parafermion state. All these
trial states are constructed according to a guiding principle called the
bilayer mapping approach, where a trial state is obtained as the
antisymmetrized projection of a bilayer quantum Hall state with interlayer
distance $d$ as a variational parameter. Under the proper understanding of the
ground-state degeneracy in the torus geometry, the $Z_4$ parafermion state can
be obtained as the antisymmetrized projection of the Halperin (330) state.
Similarly, it is proved in this work that the fermionic Haffnian state can be
obtained as the antisymmetrized projection of the Halperin (551) state. It is
shown that, while extremely accurate at sufficiently large positive Haldane
pseudopotential variation $\delta V_1^{(1)}$, the Laughlin state loses its
overlap with the exact 7/3 ground state significantly at $\delta V_1^{(1)}
\simeq 0$. At slightly negative $\delta V_1^{(1)}$, it is shown that the
PH-conjugated $Z_4$ parafermion state has a substantial overlap with the exact
7/3 ground state, which is the highest among the above four trial states.
