The author studies Hilbert's irreducibility theorem using \(s\)-integral points and Hilbert subsets. Given an integer \(s > 0\), he calls an element \(t\) of a field \(K\) to be \(s \)-integral if the set of the places \(v \in M_K\) for which \(|t |_v > 1\) is of cardinality \(\leq s \). If \(P = \{P_1, \dots, P_m\} \subset K(T) [Y]\) and \(t \in K\), then \(D_t (P)\) (respectively \(D^+_t (P))\) is the minimal (resp. maximal) degree over \(K\) of a field generated by \(m\) distinct elements \(y_1(t), \dots, y_m (t) \in \overline K\) such that \(y_i (t)\) is a root of \(P_i (t,Y)\), \(i = 1, \dots, m\); the definition extends to the points \(t = \infty\) and \(t = gen\), the point at infinity resp. the generic point of \(\mathbb{P}^1\). It is assumed that the base field \(K\) satisfies the product formula [cf. Ch. 2 of \textit{S. Lang}, Fundamentals of diophantine geometry, Springer Verlag (1983; Zbl 0528.14013)]. An improved version is obtained of \textit{V. G. Sprindzhuk}'s inequalities on the values of algebraic functions [J. Reine Angew. Math. 340, 26-52 (1983; Zbl 0497.12001)]. In a key result of the paper it is proved that if \(P_1, \dots, P_m\) are separable over \(K(T)\) and unramified above \(T = \infty\), then, for \(t\) an \(s\)-integral point of \(K\) of sufficiently large height \(h(t)\), \[ sD^+_\infty (P) D_t (P) \geq D_{gen} (P). \] In the main result about Hilbert subsets sufficient conditions are described for the existence of a polynomial \(f\) for which \(f(t)\) lies in a Hilbert subset \(H_{P_1, \dots, P_n}\) with \(t\) an \(s\)-integral point of height \(h(t) > h_2 s^2\).