  We develop combinatorial tools to study partial rigidity within the class of
minimal $\mathcal{S}$-adic subshifts. By leveraging the combinatorial data of
well-chosen Kakutani-Rokhlin partitions, we establish a necessary and
sufficient condition for partial rigidity. Additionally, we provide an explicit
expression to compute the partial rigidity rate and an associated partial
rigidity sequence. As applications, we compute the partial rigidity rate for a
variety of constant length substitution subshifts, such as the Thue-Morse
subshift, where we determine a partial rigidity rate of 2/3. We also exhibit
non-rigid substitution subshifts with partial rigidity rates arbitrarily close
to 1 and as a consequence, using products of the aforementioned substitutions,
we obtain that any number in $[0, 1]$ is the partial rigidity rate of a system.
