Transverse single-spin asymmetries to probe the transverse-spin structure of the proton have been measured for neutral pions and nonidentified charged hadrons from polarized proton-proton collisions at midrapidity and square root(s) = 200 GeV. The data cover a transverse momentum (pT) range 1.0-5.0 GeV/c for neutral pions and 0.5-5.0 GeV/c for charged hadrons, at a Feynman-x value of approximately zero. The asymmetries seen in this previously unexplored kinematic region are consistent with zero within errors of a few percent. In addition, the inclusive charged hadron cross section at midrapidity from 0.5 < pT < 7.0 GeV/c is presented and compared to next-to-leading order perturbative QCD (pQCD) calculations. Successful description of the unpolarized cross section above approximately 2 GeV/c suggests that pQCD is applicable in the interpretation of the asymmetry results in the relevant kinematic range.