We demonstrate robust single-transverse-mode light propagation in higher-order modes of a fiber, with effective area $A_{\mathrm{eff}}$ ranging from 2100 to $3200\mathrm{\mu }{\mathrm{m}}^2$ . These modes are accessed using long-period fiber gratings that enable higher-order-mode excitation over a bandwidth of $94\mathrm{mm}$ with greater than 99% of the light in the desired mode. The fiber is designed such that the effective index separation between modes is always large, hence minimizing in-fiber mode mixing and enabling light propagation over lengths as large as $12\mathrm{m}$ , with bends down to $4.5\mathrm{cm}$ radii. The modal stability increases with mode order, suggesting that $A_{\mathrm{eff}}$ of this platform is substantially scalable.