Four-dimensional (4D) Radar is a useful sensor for the detection of surrounding three-dimensional (3D) objects under various weather conditions. However, since Radar measurements could be corrupted by noise, interference, multipath, and clutter, it is necessary to employ a preprocessing algorithm to filter out invalid measurements before the object detection with neural networks. In this paper, we propose RTNH+ that is an enhanced version of RTNH, a 4D Radar object detection network, based on two novel proposed algorithms. The first algorithm is the two-level preprocessing (TLP) algorithm that produces two measurement outputs from the same 4D Radar measurements at different filtering levels. The two outputs enrich the representation of the 4D Radar measurements and enhance object detection network performance. The second is the vertical encoding (VE) algorithm that effectively encodes vertical features of the surrounding objects in the TLP outputs. We provide details of the RTNH+ and demonstrate that RTNH+ achieves significant performance improvement of 10.14% in AP(IoU=0.3,3D) and 16.12% in AP(IoU=0.5,3D) over RTNH. The codes are available at https://github.com/kaist-avelab/K-Radar.