We investigate theoretically the phase diagram of a classical Heisenberg antiferromagnet on the pyrochlore lattice perturbed by a weak second-neighbor interaction J2. The huge ground-state degeneracy of the nearest-neighbor Heisenberg spins is lifted by J2 and a magnetically ordered ground state sets in upon approaching zero temperature. We have found a new, partially ordered phase with collinear spins at finite temperatures for a ferromagnetic J2. In addition to a large nematic order parameter, this intermediate phase also exhibits a layered structure and a bond order that breaks the sublattice symmetry. Thermodynamic phase boundaries separating it from the fully disordered and magnetically ordered states scale as 1.87J2S2 and 0.26J2S2 in the limit of small J2. The phase transitions are discontinuous. We analytically examine the local stability of the collinear state and obtain a boundary T~J22/J1 in agreement with Monte Carlo simulations.