We show that for any $\varepsilon>0$ and $Δ\in\mathbb{N}$, there exists $α>0$ such that for sufficiently large $n$, every $n$-vertex graph $G$ satisfying that $δ(G)\geq\varepsilon n$ and $e(X, Y)>0$ for every pair of disjoint vertex sets $X, Y\subseteq V(G)$ of size $αn$ contains all spanning trees with maximum degree at most $Δ$. This strengthens a result of Böttcher et al.