The rings for which any polynomial with a nonzero right annihilator must have a nonzero constant right annihilator are called the right McCoy rings. This class of rings includes the duo, reversible, polynomially semicommutative, and Armendariz rings, among others. In this paper we introduce a new condition, strictly generalizing the reversible property, which still implies the McCoy condition. We call this new condition the outer McCoy property; it arises from guaranteeing annihilators in unexpected places.This outer McCoy condition is further motivated by a property of 2-primal rings, which we call the Camillo property, first noticed by Victor Camillo and the fourth author. We study the relationships between the outer McCoy property, the Camillo property, and other standard ring-theoretic conditions, with many examples delimiting their connections. For instance, we show that any ring whose set of nilpotents is closed under multiplication must satisfy the Camillo property when restricted to linear polynomials.