Magnetic field detection exploiting nitrogen-vacancy (NV) centers in diamond has gained increasing attention and development in recent years. Combining diamond NV centers to optical fibers provides a way for achieving magnetic sensors with high integration and portability. Meanwhile, new methods or techniques are urgently desired to improve the detection sensitivity of such sensors. In this paper, we present an optical-fiber magnetic sensor based on the NV ensemble in diamond, and employ the well-designed magnetic flux concentrators to enhance the sensitivity up to 12 pT/Hz1/2, an outstanding level among the diamond-integrated optical-fiber magnetic sensors. The dependence of sensitivity on the key parameters including the size and gap width of the concentrators are investigated by simulations and experiments, based on which the predictions on the further enhancement of sensitivity to fT level are presented.