We prove that, for any $t\ge 3$, there exists a constant $c=c(t)>0$ such that any $d$-regular $n$-vertex graph with the second largest eigenvalue in absolute value~$λ$ satisfying $λ\le c d^{t-1}/n^{t-2}$ contains vertex-disjoint copies of $K_t$ covering all but at most $n^{1-1/(8t^4)}$ vertices. This provides further support for the conjecture of Krivelevich, Sudakov and Szábo [\emph{Triangle factors in sparse pseudo-random graphs}, Combinatorica \textbf{24} (2004), pp.~403--426] that $(n,d,λ)$-graphs with $n\in 3\mathbb{N}$ and $λ\leq cd^{2}/n$ for a suitably small absolute constant~$c>0$ contain triangle-factors.