We demonstrate an all-fiber photon-pair source with the highest coincidence-to-accidental ratio (CAR) reported to date in the fiber-optic telecom C-band. We achieve this through careful optimization of pairproduction efficiency as well as careful characterization and minimization of all sources of background photons, including Raman generation in the nonlinear fiber, Raman generation in the single-mode fiber, and leakage of pump photons. We cool the nonlinear fiber to 4 K to eliminate most of the Raman scattering, and we reduce other noise photon counts through careful system design. This yields a CAR of 1300 at a pair generation rate of 2 kHz. This CAR is a factor of 12 higher than previously reported results in the C-band. Measured data agree well with theoretical predictions.