CoNb₂O₆ is a model system for a spin-1/2 one-dimensional (1D) transverse-field Ising magnet (TFIM) with a rather low three-dimensional (3D) Néel ordering temperature at T N = 2.95 K. We studied CoNb₂O₆ using ultrasound measurements down to 0.3 K in transverse magnetic fields applied along the b direction. Upon entering the 3D ordered state, we observe pronounced anomalies in the transverse acoustic mode c ₆₆. In particular, from 1.3 to 1.5 K and around 4.7 T, this mode reveals an almost diverging softening, which is considerably reduced at lower and higher magnetic fields. We interpret this as an influence of quantum critical fluctuations emerging from the quantum critical point (QCP) of the 1D Ising spin chains at about 4.75 T, which lies below the QCP of the 3D ordering at about 5.4 T. This is clear experimental evidence of the predicted generic phase diagram for a TFIM with superimposed 3D ordering.