Let $H$ be a nonabelian finite simple group. Huppert's conjecture asserts that if $G$ is a finite group with the same set of complex character degrees as $H$, then $G\cong H\times A$ for some abelian group $A$. Over the past two decades, several specific cases of this conjecture have been addressed. Recently, attention has shifted to the analogous conjecture for character codegrees: if $G$ has the same set of character codegrees as $H$, then $G\cong H$. Unfortunately, both problems have primarily been examined on a case-by-case basis. In this paper and the companion [HM22], we present a more unified approach to the codegree conjecture and confirm it for several families of simple groups.