We study both experimentally and using micromagnetic simulations how the spin excitation spectra of the vortex-state circular dots made of soft magnetic material change with the dot thickness $t$ in the range $t=20–80\mathrm{nm}$. It is found that in addition to higher-order gyrotropic modes which are nonuniform along the dot thickness and were observed earlier, azimuthal spin-wave modes having curled structure at the dot top and bottom faces appear in the spectrum when increasing the dot thickness. For the dot thickness $t>50\mathrm{nm}$ these “curled” modes become the lowest ones in the spin-wave excitation spectrum. It is also shown that all spin-wave modes with azimuthal index $m=\pm 1$ are hybridized with the vortex gyrotropic modes. However, while “common” azimuthal $(0,\pm 1)$ modes are hybridized with the main gyrotropic $G_0$ mode and reveal large frequency splitting of their doublet, the curled modes can be hybridized with higher-order gyrotropic modes and the doublet frequency splitting vanishes with the dot thickness increase.