Because of their ability to sustain extremely high-amplitude electromagnetic fields and transient density and field profiles, plasma optical components are being developed to amplify, compress, and condition high-power laser pulses. We recently demonstrated the potential to use a relativistic plasma aperture—produced during the interaction of a high-power laser pulse with an ultrathin foil target—to tailor the spatiotemporal properties of the intense fundamental and second-harmonic light generated [Duff et al., Sci. Rep. 10, 105 (2020)]. Herein, we explore numerically the interaction of an intense laser pulse with a preformed aperture target to generate second-harmonic laser light with higher-order spatial modes. The maximum generation efficiency is found for an aperture diameter close to the full width at half maximum of the laser focus and for a micrometer-scale target thickness. The spatial mode generated is shown to depend strongly on the polarization of the drive laser pulse, which enables changing between a linearly polarized TEM₀₁ mode and a circularly polarized Laguerre–Gaussian LG₀₁ mode. This demonstrates the use of a plasma aperture to generate intense higher-frequency light with selectable spatial mode structure.