Magnetic behavior of the pseudo-binary alloy Hf(1-x)Ta(x)Fe(2) has been studied, for which the zero-field ferromagnetic (FM) to antiferromagnetic (AFM) transition temperature is tuned near to T = 0 K. Our studies show that such composition lies around x = 0.230. Detailed magnetization studies on x = 0.225, 0.230 and 0.235 show thermomagnetic irreversibility at low temperature due to kinetic arrest of the first-order AFM-FM transition. All three compositions studied show a reentrant transition in the zero-field-cooled warming curve and non-monotonic variation of the upper critical field. The region in H-T space where these features of kinetic arrest manifest themselves increases with increasing Ta concentration.