Measurements of transverse-single-spin asymmetries (A(N)) in p + p collisions at root s = 62.4 and 200 GeV with the PHENIX detector at the Relativistic Heavy Ion Collider are presented. At midrapidity, A(N) is measured for neutral pion and eta mesons reconstructed from diphoton decay, and, at forward rapidities, neutral pions are measured using both diphotons and electromagnetic clusters. The neutral-pion measurement of A(N) at midrapidity is consistent with zero with uncertainties a factor of 20 smaller than previous publications, which will lead to improved constraints on the gluon Sivers function. At higher rapidities, where the valence quark distributions are probed, the data exhibit sizable asymmetries. In comparison with previous measurements in this kinematic region, the new data extend the kinematic coverage in root s and p(T), and it is found that the asymmetries depend only weakly on root s. The origin of the forward A(N) is presently not understood quantitatively. The extended reach to higher p(T) probes the transition between transverse momentum dependent effects at low p(T) and multiparton dynamics at high p(T).