The authors continue their study of the maximal length \(\ell(G)\) of a chain of subgroups \(G=G_ 0>G_ 1>\dots>G_ k=1\) in a finite simple group of Lie type in characteristic \(p\). In the first paper of the series [J. Algebra 132, No. 1, 174-184 (1990; Zbl 0714.20011)] they gave a precise formula in the case \(p=2\). In the second paper, with \textit{G. M. Seitz} [J. Lond. Math. Soc., II. Ser. 42, No. 1, 93-100 (1990; Zbl 0728.20018)] they considered the case \(p\) odd, and proved in particular that for given \(p\) and \(r\), a longest chain in \(G_ r(p^ m)\) goes through a Borel subgroup provided \(m\) is sufficiently large. In the present paper they prove that this lower bound on \(m\) depends on \(p\) but not essentially on \(r\). That is, they prove that there exists a positive integer \(F(p)\) such that if \(G=G_ r(p^ m)\) is a group of Lie type with \(m\geq F(p)\), then \(\ell(G)=\ell(B)+r\), where \(B\) is a Borel subgroup of \(G\).