A new procedure for improving the convergence of the reduced linear equation method for solving the single-reference (SR) coupled-cluster (CC) equations in highly degenerate cases, where the convergence of the iterative process is often destabilized by "dangerous" denominators, is discussed and implemented. Emphasis is placed on the SR CC methods involving highly excited clusters, such as the CC method with singles, doubles, triples, and quadruples (CCSDTQ) : The new algorithm is based on rearranging and quasilinearizing the nonlinear terms and correcting the energy denominators by the quasilinearized terms. The method is tested in the SR CCSDTQ calculations for a dissociating single bond.