Most efforts to detect signatures of dynamical dark energy (DE) are focused on late times, z ≲ 2, where the DE component begins to dominate the cosmic energy density. Many theoretical models involving dynamical DE exhibit a 'freezing' equation of state however, where w → -1 at late times, with a transition to a 'tracking' behaviour at earlier times (with w ≫ -1 at sufficiently high redshift). In this paper, we study whether constraints on background distance indicators from large-scale structure (LSS) surveys in the post-reionization matter-dominated regime, 2 ≲ z ≲ 6, are sensitive to this behaviour, on the basis that the DE component should remain detectable (despite being strongly subdominant) in this redshift range given sufficiently precise observations. Using phenomenological models inspired by parameter space studies of Horndeski (generalized scalar-tensor) theories, we show how existing CMB and LSS measurements constrain the DE equation of state in the matter-dominated era, and examine how forthcoming galaxy surveys and 21 cm intensity mapping instruments can improve constraints in this regime at the background level. We also find that the combination of existing CMB and LSS constraints with DESI will already come close to offering the best possible constraints on H0 using BAO/galaxy power spectrum measurements, and that either a spectroscopic follow-up of the LSST galaxy sample (e.g. MegaMapper or SpecTel) or a Stage 2/PUMA-like intensity mapping survey, both at z ≳ 2, would offer better constraints on the class of models considered here than a comparable cosmic variance-limited galaxy survey at z ≲ 1.5.