In small diameter positron emission tomography (PET) systems, the determination of the depth-of-interaction (DOI) of 511 keV gamma photons in scintillator crystals is of great importance, in order to achieve high DOI resolution with good uniformity within the entire field-of-view. In this work, we propose a new method for DOI determination, in which a single layer of LYSO crystals is read out on both ends through the use of silicon photomultipliers (SiPMs), but using wavelength-shifting fibers and a reduced number of SiPMs on one end. This design results in a simpler and less expensive readout when compared to the typical dual-ended readout method, which requires two photodetectors per crystal and corresponding readout electronics. GATE simulation of the system was carried out and experimental proof-of-concept studies were performed on a single detector cell (composed of two LYSO crystals operating in coincidence), to evaluate the amount of light detected on each side of the crystal and the achievable DOI resolution with this method, taking into account the attenuation of the light signal on the fiber side with crystal-SiPM distance. The feasibility of applying this new method in full detector rings for a small animal PET system is evaluated and discussed, considering different alternatives for position readout electronics.