The tree breadth ${\rm tb}(G)$ of a connected graph $G$ is the smallest non-negative integer $ρ$ such that $G$ has a tree decomposition whose bags all have radius at most $ρ$. We show that, given a connected graph $G$ of order $n$ and size $m$, one can construct in time $O(m\log n)$ an additive tree $O\big({\rm tb}(G)\log n\big)$-spanner of $G$, that is, a spanning subtree $T$ of $G$ in which $d_T(u,v)\leq d_G(u,v)+O\big({\rm tb}(G)\log n\big)$ for every two vertices $u$ and $v$ of $G$. This improves earlier results of Dragan and Köhler (Algorithmica 69 (2014) 884-905), who obtained a multiplicative error of the same order, and of Dragan and Abu-Ata (Theoretical Computer Science 547 (2014) 1-17), who achieved the same additive error with a collection of $O(\log n)$ trees.