Single-photon sources are of great interest because they are key elements in different promising applications of quantum technologies. Here we demonstrate a highly efficient tunable on-demand microwave single-photon source based on a transmon qubit with an intrinsic emission efficiency above 98%. The high efficiency ensures a negligible pure dephasing rate and the necessary conditions for generation of indistinguishable photons. We provide an extended discussion and analysis of the efficiency of the photon generation. To further experimentally confirm the single-photon property of the source, correlation functions of the emission field are also measured using linear detectors with a GPU-enhanced signal-processing technique. Our results experimentally demonstrate that frequency tunability and negligible pure dephasing rate can be achieved simultaneously and show that such a tunable single-photon source can be good for various practical applications in quantum communication, simulations and information processing in the microwave regime.