The authors investigate subquadrangles of order \(s\) of generalized quadrangles of order \((s,s^2)\), \(s\) odd. The case of even \(s\) is considered in [J. Comb. Theory, Ser. A 106, No. 1, 15--32 (2004; see the review above)]. Suppose that \(Q'\) is a subquadrangle of order \(s\) of a generalized quadrangle of order \((s,s^2)\), \(s\) odd. The authors show that if \(Q\) is a translation generalized quadrangle, such that the corresponding egg has a good element \(\pi\), then \(Q'\) is isomorphic to \(Q(4,s)\) and either \(Q\) is isomorphic to \(Q(5,s)\) or \(Q'\) is one of the \(s^3+s^2\) subquadrangles of order \(s\) through the line \(\pi\) of \(Q\). If \(Q\) is a dual flock GQ and \(Q'\) contains the line \(L^\ast\) corresponding with the base point \((\infty)\) of the flock GQ \(Q^D\), then \(Q^D\) is a Kantor semifield flock GQ, \(Q'\) is isomorphic to \(Q(4,s)\) and either \(Q\) is isomorphic to \(Q(5,s)\) or \(Q'\) is one of the \(s^3+s^2\) subquadrangles of order \(s\) containing the line \(L^\ast\). The authors also provide a characterization of the Kantor semifield flock GQ in terms of the net which can be defined on the base point \((\infty)\).