In a previous paper the author presented an extension of an iterative approximate orthogonalization algorithm, due to Z. Kovarik, for arbitrary rectangular matrices. In this algorithm, as Kovarik already observed in his paper, at each iteration an inversion of a symmetric and positive definite matrix is made. The dimension of this matrix equals the number of rows of the initial one, thus the inverse computation can be very expensive. In the present paper we describe an algorithm in which the above matrix inversion step is replaced by an arbitrary odd degree polynomial matrix expression. We prove that this new algorithm converges to the same matrix as the original Kovarik's method. Some numerical experiments described in the last section of the paper show us that, even for small degree polynomial expressions the convergence properties of the new algorithm are comparable with those of the original one.