Coalescence-induced droplet jumping from a round hydrophobic fiber was studied by phase-field-based hybrid lattice-Boltzmann finite-difference simulations in which the interface dynamics is handled by the finite-difference solution of the Cahn-Hilliard equation and the hydrodynamics is handled by the lattice-Boltzmann method. It was found that at a small Ohnesorge number of O(0.01), several different outcomes, including normal jumping at a positive velocity, jumping with a negative velocity, jumping after wrapping the fiber, and oscillating on the fiber, may occur after droplet coalescence, depending on the wettability of the fiber and the droplet-to-fiber radius ratio. In accordance with previous reports, normal droplet jumping from the fiber happens only when the radius ratio exceeds some critical value for a given contact angle. The critical radius ratio decreases as the fiber becomes more hydrophobic. For a given contact angle and radius ratio, it was observed that there exists an Ohnesorge number at which the jumping speed achieves a maximum value.