  Na2Co2TeO6 is a proposed approximate Kitaev magnet, yet its actual magnetic
interactions are elusive due to a lack of knowledge on the full excitation
spectrum. Here, using inelastic neutron scattering and single crystals, we
determine the system's temperature-dependent magnetic excitations over the
entire Brillouin zone. Without committing to specific models, we unveil a
distinct signature of the third-nearest-neighbor coupling in the spin waves,
which signifies the associated distance as an emerging "soft link" in the
ordered state. The presence of at least six non-overlapping spin-wave branches
is at odds with all models proposed to date. Above the ordering temperature,
persisting dynamic correlations can be described by equal-time magnetic
structure factors of a hexagonal cluster, which reveal the leading
instabilities. Our result sets definitive constraint on theoretical models for
Na2Co2TeO6 and provides new insight for the materialization of the Kitaev
model.
