We present an optically stabilized Erbium fiber frequency comb with a broad repetition rate tuning range based on a hybrid mode-locked oscillator. We lock two comb modes to narrow-linewidth reference lasers in turn to investigate the best performance of control loops. The control bandwidth of fast and slow piezoelectric transducers reaches 70 kHz, while that of pump current modulation with phase-lead compensation is extended to 32 kHz, exceeding laser intrinsic response. Eventually, simultaneous lock of both loops is realized to totally phase-stabilize the comb, which will facilitate precision dual-comb spectroscopy, laser ranging, and timing distribution. In addition, a 1.8-MHz span of the repetition rate is achieved by an automatic optical delay line that is helpful in manufacturing a secondary comb with a similar repetition rate. The oscillator is housed in a homemade temperature-controlled box with an accuracy of ±0.02 K, which not only keeps high signal-to-noise ratio of the beat notes with reference lasers, but also guarantees self-starting at the same mode-locking every time.