By adding a parameter θ in M. Jakobi et al's protocol [Phys. Rev. A 83, 022301 (2011)], we present a flexible quantum-key-distribution-based protocol for quantum private queries. We show that, by adjusting the value of θ, the average number of the key bits Alice obtains can be located on any fixed value the users wanted for any database size. And the parameter k is generally smaller (even k = 1 can be achieved) when θ < π/4, which implies lower complexity of both quantum and classical communications. Furthermore, the users can choose a smaller θ to get better database security, or a larger θ to obtain a lower probability with which Bob can correctly guess the address of Alice's query.