A search for charmonium and other new states is performed in a study of exclusive initial-state-radiation production of $D\overline{D}$ events from electron-positron annihilations at a center-of-mass energy of 10.58 GeV. The data sample corresponds to an integrated luminosity of $384\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ and was recorded by the BABAR experiment at the PEP-II storage ring. The $D\overline{D}$ mass spectrum shows clear evidence of the $\ensuremath{\psi}(3770)$ plus other structures near 3.9, 4.1, and $4.4\text{ }\text{ }\mathrm{GeV}/{c}^{2}$. No evidence for $Y(4260)\ensuremath{\rightarrow}D\overline{D}$ is observed, leading to an upper limit of $\mathcal{B}(Y(4260)\ensuremath{\rightarrow}D\overline{D})/\mathcal{B}(Y(4260)\ensuremath{\rightarrow}J/\ensuremath{\psi}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}})<1.0$ at 90% confidence level.