This paper addresses the design of gradient-based search algorithms for multivariable system estimation. In particular, the paper here considers so-called "full parametrization" approaches, and establishes that the recently developed "data-driven local coordinate" methods can be seen as a special case within a broader class of techniques that are designed to deal with rank-deficient Jacobians. This informs the design of a new algorithm that, via a strategy of dynamic Jacobian rank determination, is illustrated to offer enhanced performance.