  Conlon, Gowers, Samotij, and Schacht showed that for a given graph $H$ and a
constant $\gamma > 0$, there exists $C > 0$ such that if $p \ge Cn^{-1/m_2(H)}$
then asymptotically almost surely every spanning subgraph $G$ of the random
graph $\mathcal{G}(n,p)$ with minimum degree at least $\delta(G) \ge (1 -
1/\chi_{\mathrm{cr}}(H) + \gamma )np$ contains an $H$-packing that covers all
but at most $\gamma n$ vertices. Here, $\chi_{\mathrm{cr}}(H)$ denotes the
critical chromatic threshold, a parameter introduced by Koml\'os. We show that
this theorem can be bootstraped to obtain an $H$-packing covering all but at
most $\gamma (C/p)^{m_2(H)}$ vertices, which is strictly smaller when $p > C
n^{-1/m_2(H)}$. In the case where $H = K_3$ this answers the question of
Balogh, Lee, and Samotij. Furthermore, we give an upper bound on the size of an
$H$-packing for certain ranges of $p$.
