After a satellite has been launched, the determination of its initial orbit is one of the primary tasks to be accomplished by mission control: e.g., in order to prepare for high-precision orbit determination or orbital transfers. For this purpose, tracking measurements are acquired by ground stations, which are the basis for the determination of the unknown orbit parameters. In this paper, the orbit determination problem is formulated as a least-squares parameter estimation problem, and a newly developed multiple shooting method is presented for its solution. For the generation of initial guesses, the method incorporates a so-called analytical projection, which is specifically tailored to the available measurement data. Original tracking data of the launch and early orbit phases of the Artemis satellite and the second pair of Cluster-II satellites are used for tests of the method. Both scenarios constitute challenging benchmark problems, because the satellites were launched into orbits significantly different from the expected ones. A comparison to the widely employed single shooting method reveals that the newly developed method significantly improves the convergence behavior.