Let \(K\) be a compact subset of the complex plane \(\mathbb{C}\) such that \(\mathbb{C} - K\) is connected. Given a function \(f \in A (K)\), let \(E_n (f) : = \inf \{|f - p_n |_K; \deg p_n \leq n\}\) be the error of the best uniform approximation to \(f\) by polynomials of degree at most \(n\). Following \textit{E. B. Saff} and \textit{V. Totik} [Trans. Am. Math. Soc. 316, No. 2, 567-593 (1989; Zbl 0683.41034)], by ``near-best'' polynomial approximation of \(f\) the author means a sequence \(\{q_n\}\) with \(|f - q_n |_K \leq C \cdot E_n (f)\), where \(C\) does not depend on \(n\). Under some assumptions on the geometry of K. Saff and Totik proved that near-best approximants \(q_n\) exist which decrease faster on compact subsets of int \(K\) than \(E_n (f)\). The present author obtains lower and upper estimates of improvement of convergence of \(\{q_n\}\) on compact subsets of int \(K\).