  We propose an improved scheme to construct many-body trial wave functions for
fractional Chern insulators (FCI), using one-dimensional localized Wannier
basis. The procedure borrows from the original scheme on a continuum cylinder,
but is adapted to finite-size lattice systems with periodic boundaries. It
fixes several issues of the continuum description that made the overlap with
the exact ground states insignificant. The constructed lattice states are
translationally invariant, and have the correct degeneracy as well as the
correct relative and total momenta. Our prescription preserves the (possible)
inversion symmetry of the lattice model, and is isotropic in the limit of flat
Berry curvature. By relaxing the maximally localized hybrid Wannier orbital
prescription, we can form an orthonormal basis of states which, upon gauge
fixing, can be used in lieu of the Landau orbitals. We find that the exact
ground states of several known FCI models at nu=1/3 filling are well captured
by the lattice states constructed from the Laughlin wave function. The overlap
is higher than 0.99 in some models when the Hilbert space dimension is as large
as 3x10^4 in each total momentum sector.
