We prove that every oriented nondisk Seifert surface F for an oriented knot K in S³ is smoothly concordant to a Seifert surface F′ for a hyperbolic knot K′ of arbitrarily large volume. This gives a new and simpler proof of the result of Friedl and of Kawauchi that every knot is S-equivalent to a hyperbolic knot of arbitrarily large volume. The construction also gives a new and simpler proof of the result of Silver and Whitten and of Kawauchi that for every knot K there is a hyperbolic knot K′ of arbitrarily large volume and a map of pairs f :(S³, K′)→(S³, K) which induces an epimorphism on the knot groups. An example is given which shows that knot Floer homology is not an invariant of Seifert surface concordance. We also prove that a set of finite volume hyperbolic 3-manifolds with unbounded Haken numbers has unbounded volumes.