In this paper we describe iterative algorithms for numerical solution of linear least-squares problems. They are based on combinations between an extension, with relaxation parameters for the clasical Kaczmarz's projections method (obtained by one of the authors in a previous work) and approximate orthogonalization techniques due to Z. Kovarik. We prove that the new algorithms converge to any solution of an inconsistent and rank-defficient least-squares problem (with respect to the choice of the initial approximation), the convergence being much faster than for classical Kaczmarz - like methods. Numerical experiments on a first kind integral equation are described in the last section of the paper.