A search is made for a vector-like $T$ quark decaying into a Higgs boson and a top quark in 13 TeV proton-proton collisions using the ATLAS detector at the Large Hadron Collider with a data sample corresponding to an integrated luminosity of 139$^{-1}$ fb. The all-hadronic decay modes $H \rightarrow b\bar{b}$ and $t \rightarrow bW \rightarrow bq\bar{q}' $ are reconstructed as large-radius jets and identified using tagging algorithms. Improvements in background estimation, signal discrimination, and a larger data sample, contribute to an improvement in sensitivity over previous all-hadronic searches. No significant excess is observed above the background, so limits are set on the production cross-section of a singlet $T$ quark at 95\% confidence level, depending on the mass, $m_{T}$, and coupling, $\kappa_{T}$, of the vector-like $T$ quark to Standard Model particles. This search targets a mass range between 1.0 to 2.3 TeV, and a coupling value between 0.1 to 1.6, expanding the phase space of previous searches. In the considered mass range, the upper limit on the allowed coupling values increases with $m_{T}$ from a minimum value of 0.35 for $1.07 < m_{T} < 1.4$ TeV up to 1.6 for $m_{T} = 2.3$ TeV.