  We theoretically analyze a discrete Schr\"odinger chain with hopping to the
first and second neighbors, as can be realized with zigzag arrangements of
optical waveguides or lattice sites for cold atoms. Already at moderate values,
the second-neighbor hopping has a strong impact on the band structure, leading
to the emergence of a new extremum located inside the band, accompanied by a
van Hove singularity in the density of states. The energy band is then divided
into a subcritical regime with the usual unique correspondence between wave
number and energy of the travelling waves, and a supercritical regime, in which
waves of different wave number are degenerate in energy. We study the
consequences of these features in a scattering setup, introducing a defect that
locally breaks the translational invariance. The notion of a local probability
current is generalized beyond the nearest-neighbor approximation and bound
states with energies outside the band are discussed. At subcritical energies
inside the band, an evanescent mode coexists with the travelling plane wave,
giving rise to resonance phenomena in scattering. At weak coupling to the
defect, we identify a prototypical Fano-Feshbach resonance of tunable shape and
provide analytical expressions for its profile parameters. At supercritical
energies, we observe coupling of the degenerate travelling waves, leading to an
intricate wave packet fragmentation dynamics. The corresponding branching
ratios are analyzed.
