The Ultracold Neutron Asymmetry (UCNA) experiment was designed to measure the $beta$-decay asymmetry parameter, $A_0$, for free neutron decay. In the experiment, polarized ultracold neutrons are transported into a decay trap, and their $beta$-decay electrons are detected with $approx 4pi$ 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 $beta$-decay rate. In this work, we determine $b_{n}$ from the energy dependence of $A_0$ 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 $b_{n}$ measurement using the weighted average of our asymmetry dataset fits, to obtain $b_{n} = 0.066 pm 0.041_{text{stat}} pm 0.024_{text{syst}}$ which corresponds to a limit of $-0.012 < b_{n} < 0.144$ at the 90% confidence level.
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