The production of ultracold molecules with their rich internal structure is currently attracting considerable interest(1-4). For future experiments, it will be important to efficiently transfer these molecules from their initial internal quantum state at production to other quantum states of interest. Transfer tools such as optical Raman schemes(5,6), radiofrequency transitions (see, for example, ref. 7) or magnetic field ramping(8,9) exist, but are either technically involved or limited in their applicability. Here, we demonstrate a simple, highly efficient hybrid transfer method that overcomes a number of the previous limitations. The scheme is based on magnetically tuned mixing of two neighbouring molecular levels, which enables otherwise forbidden radiofrequency transitions between them. By repeating this process at various magnetic fields, molecules can be successively transported through a large manifold of quantum states. Applying nine transfers, we convert very weakly bound Feshbach molecules to a much more deeply bound level with a binding energy corresponding to 3.6 GHz. As an important spin-off of our experiments, we demonstrate a high-precision spectroscopy method for investigating level crossings.