Most known results on avoiding the occur-check are based on the notion of "not subject to occurcheck" (NSTO). It means that unification is performed only on such pairs of atoms for which the occur-check never succeeds in any run of a nondeterministic unification algorithm. Here we show that this requirement is too strong. We show how to weaken it, and present some related sufficient conditions under which the occur-check may be safely omitted. We show examples for which the proposed approach provides more general results than the approaches based on well-moded and nicely moded programs (this includes cases to which the latter approaches are inapplicable).