Since this result is derived by Lieberman, Rosicky, and Vasey as a corollary of some more sophisticated constructions with more sophisticated goals, I think it might be worth "compiling out" the proof here. It turns out to be not so bad. Let $\mathcal C$ be a locally $\lambda$-presentable category, and recall the following facts:
For any $C \in \mathcal C$, if $\mathrm{rk}(C) > \lambda$, then $\mathrm{rk}(C) = \kappa^+$ is a successor. (Proof hint: $\kappa$ is the smallest cardinal such that $C$ is a retract of a $\kappa$-sized colimit of $\lambda$-presentable objects.)
$\mathcal C$ is locally $\kappa$-presentable for any regular $\kappa \geq \lambda$.
Theorem [Lieberman, Rosicky, and Vasey] Let $\mathcal C$ be a locally $\lambda$-presentable category and let $C \in \mathcal C$ with $\mathrm{rk}(C) = \kappa^+ > \lambda$. Then $C$ is almost filtrable.
Proof: First consider the case where $\kappa$ is regular. By (2), $\mathcal C$ is locally $\kappa$-presentable. Write $C$ as a $\kappa$-filtered colimit of $\kappa$-presentable objects. It's not hard to see that $C$ is a retract of some $\kappa$-filtered, $\kappa^+$-small subdiagram, and it's straightforward to show that any $\kappa$-filtered, $\kappa^+$-small diagram admits a cofinal chain.
Now consider the case where $\kappa$ is singular. Write $C = \varinjlim_{i \in I} C_i$ as a $\lambda$-directed colimit of $\lambda$-presentable objects. Then $C$ is a retract of the colimit of a $\kappa^+$-small subdiagram, so we may assume without loss of generality that the diagram $I$ is of cardinality $\kappa$. We may write $I = \cup_{\alpha < \mathrm{cf}(\kappa)} I_\alpha$ as the union of a $\mathrm{cf}(\kappa)$-sized increasing chain of subdiagrams of cardinality $|I_\alpha| <\kappa$. Setting $C_\alpha = \varinjlim_{i \in I_\alpha} C_i$, we have $C = \varinjlim_{\alpha < \mathrm{cf}(\kappa)} C_\alpha$, yielding the desired filtration.
(In fact it's not strictly necessary to separate these two cases, but I think it clarifies things.)