Nucleon magnetic moments display a rich nonanalytic dependence on the quark mass in both quenched and full QCD. They provide a forum for a detailed examination of the connection between quenched and full QCD made possible through the formalism of finite-range regularized chiral effective field theory. By defining meson-cloud and core contributions through the careful selection of a regularization scale, one can correct the meson cloud of quenched QCD to make full QCD predictions. Whereas past success is based on unquenching the leading-order loop contributions, here we extend and test the formalism including next-to-leading-order loop contributions. We discuss the subtleties associated with working at next-to-leading order and illustrate the role of higher-order corrections.