The rapid acquisition of high-resolution spatial distribution of soil organic matter (SOM) at the field scale is essential for precision agriculture. The UAV imaging hyperspectral technology, with its high spatial resolution and timeliness, can fill the research gap between ground-based monitoring and remote sensing. This study aimed to test the feasibility of using UAV hyperspectral data (400–1000 nm) with a small-sized calibration sample set for mapping SOM at a 1 m resolution in typical low-relief black soil areas of Northeast China. The experiment was conducted in an approximately 20 ha field. For calibration, 20 samples were collected using a 100 × 100 m grid sampling strategy, while 20 samples were randomly collected for independent validation. UAV captured hyperspectral images with a spatial resolution of 0.05 × 0.05 m. The extracted spectra within every 1 × 1 m were then averaged to represent the spectra of that grid; this procedure was also performed across the whole field. Upon applying various spectral pretreatments, including absorbance conversion, multiple scattering correction, Savitzky–Golay smoothing filtering, and first-order differentiation, the absolute maximum values of the correlation coefficients of the spectra for SOM increased from 0.41 to 0.58. Importance analysis from the optimal random forest (RF) model showed that the characterized bands of SOM were located in the 450–600 and 750–900 nm regions. When the RF model was used, the UAV hyperspectra data (UAV-RF) were able to successfully predict SOM, with an R2 of 0.53 and RMSE of 1.48 g kg−1. The prediction accuracy was then compared with that obtained using ordinary kriging (OK) and the RF model based on proximal sensing (PS-RF) with the same number of calibration samples. However, the OK method failed to predict the SOM accuracy (RMSE = 2.17 g kg−1; R2 = 0.02) due to a low sampling density. The semi-covariance function was unable to describe the spatial variability of SOM effectively. When the sampling density was increased to 50 × 50 m, OK successfully predicted SOM, with RMSE = 1.37 g kg−1 and R2 = 0.59, and its results were comparable to those of UAV-RF. The prediction accuracy of PS-RF was generally consistent with that of UAV-RF, with RMSE values of 1.41 g kg−1 and 1.48 g kg−1 and R2 values of 0.57 and 0.53, respectively, which indicated that SOM prediction based on UAV-RF is feasible. Additionally, compared with the PS platforms, the UAV hyperspectral technology could simultaneously provide spectral information of tens or even hundreds of continuous bands and spatial information at the same time. This study provides a reference for further research and development of UAV hyperspectral techniques for fine-scale SOM mapping using a small number of samples.