Power scaling based on traditional ytterbium-doped fibers (YDFs) is limited by optical nonlinear effects and transverse mode instability (TMI) in high-power fiber lasers. Here, we propose a novel long tapered fiber with a constant cladding and tapered core (CCTC) along its axis direction. The tapered-core region of the fiber is designed to enhance the stimulated Raman scattering (SRS) threshold and suppress higher-order mode resonance in the laser cavity. The CCTC YDF was fabricated successfully with a modified chemical vapor deposition (MCVD) method combined with solution doping technology, which has a cladding diameter of 400 μm and a varying core with a diameter of ~24 μm at both ends and ~31 μm in the middle. To test the performance of the CCTC fiber during high-power operation, an all-fiber laser oscillator based on a CCTC YDF was investigated experimentally. As a result, a maximum output power of 3.42 kW was achieved with an optical-to-optical efficiency of 55.2%, although the TMI effect was observed at an output power of ~3.12 kW. The measured beam quality (M2 factor) was ~1.7, and no sign of the Raman component was observed in the spectrum. We believe that CCTC YDF has great potential to simultaneously mitigate the SRS and TMI effects, and further power scaling is promising by optimizing the structure of the YDF.