The honeycomb magnet Na2Co2TeO6 has recently been argued to realize an approximate hidden SU(2) symmetry that can be understood by means of a duality transformation. Using large-scale classical Monte Carlo simulations, we study the finite-temperature phase diagram of the pertinent Heisenberg-Kitaev-Γ-Γ′ model near the hidden-SU(2)-symmetric point, in the presence of a six-spin ring exchange perturbation. At low temperatures, the model features collinear single-q zigzag and noncollinear triple-q ground states, depending on the sign of the ring exchange coupling. We show that in the vicinity of the hidden-SU(2)-symmetric point, the magnetic long-range orders melt in two stages. The corresponding finite-temperature transitions are continuous and fall into two-dimensional (2D) Ising and 2D Potts universality classes, respectively. The two fluctuation-induced phases at intermediate temperatures spontaneously break spin rotational and lattice translational symmetries, respectively, but both leave time-reversal symmetry intact. They are characterized by finite expectation values of a real, symmetric, traceless, second-rank tensor and are naturally understood as vestigial orders of the underlying magnetic states. We identify these vestigial orders as Z3 spin nematic and Z4 spin current density wave phases, respectively. For increasing ring exchange perturbations, the width of the vestigial phases decreases, eventually giving rise to a direct first-order transition from the magnetically ordered phase to the disordered paramagnet. We propose the Z4 spin current density wave phase, which is the vestigial phase of the primary triple-q magnetic order, as a natural candidate for the paramagnetic 2D long-range-ordered state observed in Na2Co2TeO6 in a small window above the antiferromagnetic ordering temperature.