A variant of the classical Koebe-logarithm osculation algorithm for conformal mapping is obtained by inserting a hyperbolic sine at an intermediate step. The modulus of convergence is calculated, and numerical experiments are reported, in particular in comparison with the method of Grassmann [E. Grassmann (1979). Numerical experiments with a method of successive approximation for conformal mapping. J. Applied Mathematics and Physics, 30, 873-884.]. Either procedure may work better, depending upon the domain. Further numerical examples show how the osculation method can be coupled to faster converging algorithms (which tend to work best for nearly-circular domains), thus making feasible computations which would not be accessible by either method alone.