The Ultracold Neutron Asymmetry (UCNA) experiment was designed to measure the β-decay asymmetry parameter, A₀, for free neutron decay. In the experiment, polarized ultracold neutrons are transported into a decay trap, and their β-decay electrons are detected with ≈4π acceptance into two detector packages which provide position and energy reconstruction. The experiment also has sensitivity to b_n, the Fierz interference term in the neutron β-decay rate. In this work, we determine bn from the energy dependence of A₀ using the data taken during the UCNA 2011–2013 run. In addition, we present the same type of analysis using the earlier 2010 A dataset. Motivated by improved statistics and comparable systematic errors compared to the 2010 data-taking run, we present a new bn measurement using the weighted average of our asymmetry dataset fits, to obtain b_n = 0.066±0.041_stat±0.024_syst which corresponds to a limit of −0.012 < b_n <0.144 at the 90% confidence level.