Superconducting qubits with in-situ tunable properties are important for constructing a quantum computer. Qubit tunability, however, often comes at the expense of increased noise sensitivity. Here, we propose a flux-tunable superconducting qubit that minimizes the dephasing due to magnetic flux noise by engineering controllable flux sweet spots at frequencies of interest. This is realized by using a SQUID with asymmetric Josephson junctions shunted by a superinductor formed from an array of junctions. Taking into account correlated global and local noises, it is possible to improve dephasing time by several orders of magnitude. The proposed qubit can be used to realize fast, high-fidelity two-qubit gates in large-scale quantum processors, a key ingredient for implementing fault-tolerant quantum computers.