Gauge fixing is applied in several contexts within lattice field theory calculations, for example to give meaning to gauge-variant observables used in RI-MOM renormalization schemes [1], as a computational trick to replace gauge-invariant operators with cheaper gauge-variant operators [2], or as inputs for comparison to phenomenological models [3, 4]. Recently, gauge-variant operators have also been used for contour deformations to reduce statistical noise [5-7]. In these contexts, the choice of gauge-fixing scheme can affect the efficiency of the calculation, and it may be desirable to systematically explore options for the scheme. Two kinds of gauge-fixing schemes are commonly used: gauge fixing by functional minimization (e.g. Landau and Coulomb gauge) or gauge fixing a maximal tree of links to the identity. Our work makes several contributions in this context. First, we parameterize a continuous family of gauge-fixing schemes that include the former as special cases. Second, we derive the gradients with respect to these parameters of an arbitrary loss function computed from gauge-fixed configurations, which can be used for gradient-based optimization within the family. Finally, we discuss the restriction of this method to a subfamily consisting of maximal trees alone, addressing the discrete nature of this space by introducing a temperature regulator, and demonstrate the effectiveness of this approach in solving two regression problems.