The theory of quantum quenches in near-critical one-dimensional systems formulated in Delfino (2014 J. Phys. A: Math. Theor. 402001) yields analytic predictions for the dynamics, unveils a qualitative difference between noninteracting and interacting systems, with undamped oscillations of one-point functions occurring only in the latter case, and explains the presence and role of different time scales. Here we examine additional aspects, determining in particular the relaxation value of one-point functions for small quenches. For a class of quenches we relate this value to the scaling dimensions of the operators. We argue that the E8 spectrum of the Ising chain can be more accessible through a quench than at equilibrium, while for a quench of the plane anisotropy in the XYZ chain we obtain that the one-point function of the quench operator switches from damped to undamped oscillations at ∆ = 1 2