A new method has been developed for obtaining multiple solutions of the classical angles-only initial-orbit-determination problem. The method operates by a higher-order Newton correction of the assumed values for two of the unknown ranges, with the author's universal Lambert algorithm at the heart of the procedure. The observations are permitted to span several revolutions when the orbit is elliptic, and the method is free of artificial singularities in the configuration of observers and sight-lines; thus it has been successfully used with some test problems that would not be solvable by existing methods.