Amplitude mode is collective excitation emerging from frozen lattice distortions below the charge-density-wave (CDW) transition temperature T_CDW and relates to the order parameter. Generally, the amplitude mode is non-polar (symmetry-even) and does not interact with incoming infrared photons. However, if the amplitude mode is polar (symmetry-odd), it can potentially couple with incoming photons, thus forming a coupled phonon–polariton quasiparticle that can travel with light-like speed beyond the optically excited region. Here, we present the amplitude mode dynamics far beyond the optically excited depth of ∼150 nm in the CDW phase of ∼10-μm-thick single-crystal EuTe₄ using time-resolved x-ray diffraction. The observed oscillations of the CDW peak, triggered by photoexcitation, occur at the amplitude mode frequency ω_AM. However, the underdamped oscillations and their propagation beyond the optically excited depth are at odds with the observation of the overdamped nature of the amplitude mode measured using meV-resolution inelastic x-ray scattering and polarized Raman scattering. The ω_AM is found to decrease with increasing fluence owing to a rise in the sample temperature, which is independently confirmed using polarized Raman scattering and ab-initio molecular dynamics simulations. We rationalize the above observations by explicitly calculating two coupled quasiparticles—phonon–polariton and exciton–polariton. Our data and simulations cannot conclusively confirm or rule out the one but point toward the likely origin from propagating phonon–polariton. The observed non-local behavior of amplitude mode thus provides an opportunity to engineer material properties at a substantially faster time scale with optical pulses.