GRANDProto300 is the planned 300-antenna pathfinder array of the Giant Radio Array for Neutrino Detection (GRAND), of which the first 100 detection units have been already produced. Its main goal is to demonstrate the viability of the detection of the radio emission from air showers initiated by inclined ultra-high-energy cosmic rays with energies of $10^{16.5}$ to $10^{18.5}$ eV, covering the purported transition region from their Galactic to extragalactic origin. The front-end readout system of each detection unit of GRANDProto300 processes signals from the radio antenna and the particle detector, generates the first-level trigger, and communicates with a central processing station. Based on earlier designs, we have built the first prototype of this system using two development boards and one self-designed front-end board. We present our new design that is improved and more economical than the earlier one, as well as test results and prospects for future work.