The classic model of a one-dimensional thermoelastic rod suspended between a hot and cold wall is revisited. In this model, the rod is held in place at the cold end, while at the hot end it is allowed to separate from or make contact with the wall. When the model includes the contact and gap dependent thermal boundary condition known as the Barber condition it serves to illustrate the well-known thermoelastic contact instability. All previous studies of this instability have focused upon the symmetric case where, as a control parameter is varied, the system undergoes a pitchfork bifurcation. That is, a new pair of linearly stable steady-state solutions bifurcate symmetrically from a previously unique solution which has changed from stable to unstable. Here, it is shown that this behavior is not generic. Rather, for typical contact resistance functions, a fold bifurcation is encountered. This represents a generic unfolding of the classic pitchfork bifurcation and contains the pitchfork as a special case.