We report on a quadratically convergent self-consistent field (QC-SCF) algorithm for the spin-projected unrestricted Hartree-Fock (SUHF) to mitigate the slow convergence of SUHF due to the presence of small eigenvalues in the orbital Hessian matrix. The new QC-SCF is robust and stable, allowing us to obtain the SUHF solutions very quickly. To demonstrate the applicability of the method, we present results for test systems with abundant non-dynamic correlation in comparison with the Roothaan repeated diagonalization, Pople extrapolation, and direct inversion of iterative subspace.