Let G be a simple linear algebraic group defined over an algebraically closed field of characteristic p >= 0 and let phi be a nontrivial p-restricted irreducible representation of G. Let T be a maximal torus of G and s epsilon T. We say that s is Ad-regular if alpha(s) not equal beta(s) for all distinct T-roots alpha and beta of G. Our main result states that if all but one of the eigenvalues of phi(s) are of multiplicity 1 then, with a few specified exceptions, s is Ad-regular. This can be viewed as an extension of our earlier work, in which we show that, under the same hypotheses, either s is regular or G is a classical group and phi is "essentially" (a twist of) the natural representation of G.