In optimal designs of infinite impulse response (IIR) filters, a major challenge is how to choose proper stability constraints to insure the stability of the designed filters. A method was proposed in [12] to iteratively and sequentially convert the minimax design of an IIR filter to a sequence of minimax designs of sub-filters with second-order denominator, for which the necessary and sufficient stability triangles can be easily applied as their stability constraints. A defect of the method, however, is its long design time, mainly because each inner iteration may need several calls of the core convex subproblem solver. The method is improved in this paper by firstly converting the non-convex minimax designs of the sub-filters into convex ones with the first-order Taylor expansion based Gauss-Newton strategy and then reducing the number of convex sub-filter design problems to one in each inner iteration. Design examples show that the improved method may obtain comparable minimax error with much less iterations and much less design time.