The out-of-equilibrium dielectric response of a glass sample after the application of a strong dc field at low temperatures cannot solely be understood by interacting two-level tunneling systems [S. Ludwig and D. D. Osheroff, Phys. Rev. Lett. 91, 105501 (2003)]. The usual picture of a particle tunneling between two close potential wells has to be extended to include at least a third well. We investigate such a three-well system. Particles trapped in the third well after the dc field application result in an additional contribution to the out-of-equilibrium response. Analyzing the experiments with regard to our theoretical model we find that the energy of the third well has a minimal value ${E}_{c}$ larger than ${k}_{B}T$ for the experimental investigated temperatures. Additionally, the tunneling frequencies in between the double well and between the third well and the double well are correlated. Such a correlation is well known for tunneling defects in crystals and we speculate that crystals doped with tunneling defects show glassy low-temperature behavior if heavily strained. Therefore, we want to encourage experiments toward that direction.