  We show that numerical linked cluster expansions (NLCEs) based on
sufficiently large building blocks allow one to obtain accurate low-temperature
results for the thermodynamic properties of spin lattice models with continuous
disorder distributions. Specifically, we show that such results can be obtained
computing the disorder averages in the NLCE clusters before calculating their
weights. We provide a proof of concept using three different NLCEs based on L,
square, and rectangle building blocks. We consider both classical (Ising) and
quantum (Heisenberg) spin-$\frac{1}{2}$ models and show that convergence can be
achieved down to temperatures that are up to two orders of magnitude lower than
the relevant energy scale in the model. Additionally, we provide evidence that
in one dimension one can obtain accurate results for observables such as the
energy down to their ground-state values.
