[en] Emittance compensation using the static axial magnetic field from a solenoid surrounding an rf photoinjector has been used to reduce the rms emittance of the electron beam by up to an order of magnitude, for photoinjectors ranging from 433 MHz to 8 GHz. The residual emittance after standard solenoidal compensation depends primarily on how linear the space-charge force is along the electron bunch in terms of an axial Taylor expansion, and is typically a strong function of both solenoid position and focusing strength. In this paper, the authors investigate the concept of second-order emittance compensation using the combination of a solenoid and radial rf focusing. They numerically demonstrate that (1) lower residual emittances are possible if second-order compensation is introduced by adding radially focusing rf forces in the first photoinjector cavity near the cathode, (2) the residual emittance is less sensitive to solenoid position, (3) the residual emittance is less sensitive to solenoid strength, and (4) the optimal solenoid position is further from the photoinjector cathode, leading to less stringent design requirements, especially at high frequencies